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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqdrng 20601 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng0 20602 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng1 20603 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngneg 20604 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngdiv 20605 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds        /r

Theoremqabvle 20606 By using induction on , we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremqabvexp 20607 Induct the product rule abvmul 15429 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostthlem1 20608* Lemma for ostth 20620. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostthlem2 20609* Lemma for ostth 20620. Refine ostthlem1 20608 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
flds        AbsVal

Theoremqabsabv 20610 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabv 20611* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvf 20612* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvcxp 20613* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostth1 20614* - Lemma for ostth 20620: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If is equal to on the primes, then by complete induction and the multiplicative property abvmul 15429 of the absolute value, is equal to on all the integers, and ostthlem1 20608 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem2 20615* Lemma for ostth2 20618. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem3 20616* Lemma for ostth2 20618. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem4 20617* Lemma for ostth2 20618. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2 20618* - Lemma for ostth 20620: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth3 20619* - Lemma for ostth 20620: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth 20620* Ostrowski's theorem, which classifies all absolute values on . Any such absolute value must either be the trivial absolute value , a constant exponent times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

PART 14  MISCELLANEA

14.1  Definitional Examples

Theoremex-or 20621 Example for df-or 361. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

Theoremex-an 20622 Example for df-an 362. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

Theoremex-dif 20623 Example for df-dif 3081. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-un 20624 Example for df-un 3083. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-in 20625 Example for df-in 3085. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-uni 20626 Example for df-uni 3728. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremex-ss 20627 Example for df-ss 3089. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-pss 20628 Example for df-pss 3091. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-pw 20629 Example for df-pw 3532. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremex-pr 20630 Example for df-pr 3551. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-br 20631 Example for df-br 3921. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-opab 20632* Example for df-opab 3975. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-eprel 20633 Example for df-eprel 4198. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-id 20634 Example for df-id 4202. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-po 20635 Example for df-po 4207. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-xp 20636 Example for df-xp 4594. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-cnv 20637 Example for df-cnv 4596. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-co 20638 Example for df-co 4597. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-dm 20639 Example for df-dm 4598. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-rn 20640 Example for df-rn 4599. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-res 20641 Example for df-res 4600. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-ima 20642 Example for df-ima 4601. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-fv 20643 Example for df-fv 4608. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-1st 20644 Example for df-1st 5974. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-2nd 20645 Example for df-2nd 5975. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theorem1kp2ke3k 20646 Example for df-dec 10004, 1000 + 2000 = 3000.

This proof disproves (by counter-example) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with , commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 10004 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

;;; ;;; ;;;

Theoremex-fl 20647 Example for df-fl 10803. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-dvds 20648 3 divides into 6. A demonstration of df-divides 12406. (Contributed by David A. Wheeler, 19-May-2015.)

14.2  Natural deduction examples

These are examples of how natural deduction rules can be applied in metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 4 and http://us.metamath.org/mpegif/mmnatded.html.

Theoremex-natded5.2 20649 Theorem 5.2 of [Laboreo] p. 15, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15 Given \$e.
22 Given \$e.
31 Given \$e.
43 E 2,3 mpd 16, the MPE equivalent of E, 1,2
54 I 4,3 jca 520, the MPE equivalent of I, 3,1
66 E 1,5 mpd 16, the MPE equivalent of E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 20650. A proof without context is shown in ex-natded5.2i 20651. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.2-2 20650 A more efficient proof of Theorem 5.2 of [Laboreo] p. 15. Compare with ex-natded5.2 20649 and ex-natded5.2i 20651. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.2i 20651 The same as ex-natded5.2 20649 and ex-natded5.2-2 20650 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.3 20652 Theorem 5.3 of [Laboreo] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 20653. A proof without context is shown in ex-natded5.3i 20654. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 Given \$e; adantr 453 to move it into the ND hypothesis
25;6 Given \$e; adantr 453 to move it into the ND hypothesis
31 ...| ND hypothesis assumption simpr 449, to access the new assumption
44 ... E 1,3 mpd 16, the MPE equivalent of E, 1.3. adantr 453 was used to transform its dependency (we could also use imp 420 to get this directly from 1)
57 ... E 2,4 mpd 16, the MPE equivalent of E, 4,6. adantr 453 was used to transform its dependency
68 ... I 4,5 jca 520, the MPE equivalent of I, 4,7
79 I 3,6 ex 425, the MPE equivalent of I, 8

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.3-2 20653 A more efficient proof of Theorem 5.3 of [Laboreo] p. 16. Compare with ex-natded5.3 20652 and ex-natded5.3i 20654. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.3i 20654 The same as ex-natded5.3 20652 and ex-natded5.3-2 20653 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.5 20655 Theorem 5.5 of [Laboreo] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 Given \$e; adantr 453 to move it into the ND hypothesis
25 Given \$e; we'll use adantr 453 to move it into the ND hypothesis
31 ...| ND hypothesis assumption simpr 449
44 ... E 1,3 mpd 16 1,3
56 ... IT 2 adantr 453 5
67 I 3,4,5 pm2.65da 562 4,6

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is mtod 170; a proof without context is shown in mto 169.

(Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Theoremex-natded5.7 20656 Theorem 5.7 of [Laboreo] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 20657. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16 Given \$e. No need for adantr 453 because we do not move this into an ND hypothesis
21 ...| ND hypothesis assumption (new scope) simpr 449
32 ... IL 2 orcd 383, the MPE equivalent of IL, 1
43 ...| ND hypothesis assumption (new scope) simpr 449
54 ... EL 4 simpld 447, the MPE equivalent of EL, 3
66 ... IR 5 olcd 384, the MPE equivalent of IR, 4
77 E 1,3,6 mpjaodan 764, the MPE equivalent of E, 2,5,6

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.7-2 20657 A more efficient proof of Theorem 5.7 of [Laboreo] p. 19. Compare with ex-natded5.7 20656. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.8 20658 Theorem 5.8 of [Laboreo] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11 Given \$e; adantr 453 to move it into the ND hypothesis
23;4 Given \$e; adantr 453 to move it into the ND hypothesis
37;8 Given \$e; adantr 453 to move it into the ND hypothesis
41;2 Given \$e. adantr 453 to move it into the ND hypothesis
56 ...| ND Hypothesis/Assumption simpr 449. New ND hypothesis scope, each reference outside the scope must change antedent to .
69 ... I 5,3 jca 520 (I), 6,8 (adantr 453 to bring in scope)
75 ... E 1,6 mpd 16 (E), 2,4
812 ... E 2,4 mpd 16 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 I 5,7,8 pm2.65da 562 (I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 20659.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.8-2 20659 A more efficient proof of Theorem 5.8 of [Laboreo] p. 20. For a longer line-by-line translation, see ex-natded5.8 20658. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.13 20660 Theorem 5.13 of [Laboreo] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 20661. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115 Given \$e.
2;32 Given \$e. adantr 453 to move it into the ND hypothesis
39 Given \$e. ad2antrr 709 to move it into the ND sub-hypothesis
41 ...| ND hypothesis assumption simpr 449
54 ... E 2,4 mpd 16 1,3
65 ... I 5 orcd 383 4
76 ...| ND hypothesis assumption simpr 449
88 ... ...| (sub) ND hypothesis assumption simpr 449
911 ... ... E 3,8 mpd 16 8,10
107 ... ... IT 7 adantr 453 6
1112 ... I 8,9,10 pm2.65da 562 7,11
1213 ... E 11 notnotrd 107 12
1314 ... I 12 olcd 384 13
1416 E 1,6,13 mpjaodan 764 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.13-2 20661 A more efficient proof of Theorem 5.13 of [Laboreo] p. 20. Compare with ex-natded5.13 20660. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded9.20 20662 Theorem 9.20 of [Laboreo] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11 Given \$e
22 EL 1 simpld 447 1
311 ER 1 simprd 451 1
44 ...| ND hypothesis assumption simpr 449
55 ... I 2,4 jca 520 3,4
66 ... IR 5 orcd 383 5
78 ...| ND hypothesis assumption simpr 449
89 ... I 2,7 jca 520 7,8
910 ... IL 8 olcd 384 9
1012 E 3,6,9 mpjaodan 764 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 453; simpr 449 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 20663. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Theoremex-natded9.20-2 20663 A more efficient proof of Theorem 9.20 of [Laboreo] p. 45. Compare with ex-natded9.20 20662. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Theoremex-natded9.26 20664* Theorem 9.26 of [Laboreo] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that is bound in the conclusion. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13 Given \$e.
26 ...| ND hypothesis assumption simpr 449. Later statements will have this scope.
37;5,4 ... E 2,y a4sbcd 2934 (E), 5,6. To use it we need a1i 12 and vex 2730. This could be immediately done with 19.21bi 1774, but we want to show the general approach for substitution.
412;8,9,10,11 ... I 3,a a4esbcd 3003 (I), 11. To use it we need sylibr 205, which in turn requires sylib 190 and two uses of sbcid 2937. This could be more immediately done using 19.8a 1758, but we want to show the general approach for substitution.
513;1,2 E 1,2,4,a exlimdd 1933 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1629 and nfe1 1566 (MPE# 1,2)
614 I 5 alrimiv 2012 (I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps).

Note that in the original proof, has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 20665.

(Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.)

Theoremex-natded9.26-2 20665* A more efficient proof of Theorem 9.26 of [Laboreo] p. 45. Compare with ex-natded9.26 20664. (Contributed by Mario Carneiro, 9-Feb-2017.)

14.3  Humor

14.3.1  April Fool's theorem

Theoremavril1 20666 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid germanus dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at http://us.metamath.org/mpegif/mmnotes.txt, under the 1-Apr-2006 entry.

Theorem2bornot2b 20667 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremhelloworld 20668 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem1p1e2apr1 20669 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeqid1 20670 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem1div0apr 20671 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

14.4  (Future - to be reviewed and classified)

14.4.1  Planar incidence geometry

Syntaxcplig 20672 Extend class notation with the class of all planar incidence geometries.

Definitiondf-plig 20673* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)

Theoremisplig 20674* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)

Theoremtncp 20675* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)

Theoremlpni 20676* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)

14.4.2  Algebra preliminaries

Syntaxcrpm 20677 Ring primes.
RPrime

Definitiondf-rprm 20678* Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 12661. (Contributed by Mario Carneiro, 17-Feb-2015.)
RPrime Unit r

14.4.3  Transitive closure

Syntaxctcl 20679 Extend class notation to include the transitive closure symbol.

Syntaxcrtcl 20680 Extend class notation with transitive closure.

Definitiondf-trcl 20681* Transitive closure of a relation. Experimental. (Contributed by FL, 27-Jun-2011.)

Definitiondf-rtrcl 20682* Reflexive-transitive closure of a relation. Experimental. (Contributed by FL, 27-Jun-2011.)

PART 15  DEPRECATED SECTIONS

15.1  Additional material on Group theory

15.1.1  Definitions and basic properties for groups

Syntaxcgr 20683 Extend class notation with the class of all group operations.

Syntaxcgi 20684 Extend class notation with a function mapping a group operation to the group's identity element.
GId

Syntaxcgn 20685 Extend class notation with a function mapping a group operation to the inverse function for the group.

Syntaxcgs 20686 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.

Syntaxcgx 20687 Extend class notation with a function mapping a group operation to the power operation for the group.

Definitiondf-grpo 20688* Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Definitiondf-gid 20689* Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId

Definitiondf-ginv 20690* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
GId

Definitiondf-gdiv 20691* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-gx 20692* Define a function that maps a group operation to the group's power operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
GId

Theoremisgrpo 20693* The predicate "is a group operation." Note that is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremisgrpo2 20694* The predicate "is a group operation." (Contributed by NM, 23-Oct-2012.) (New usage is discouraged.)

Theoremisgrpoi 20695* Properties that determine a group operation. Read as . (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremgrpofo 20696 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)

Theoremgrpocl 20697 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

Theoremgrpolidinv 20698* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Theoremgrpon0 20699 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)

Theoremgrpoass 20700 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)

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