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Mirrors > Home > MPE Home > Th. List > 2p2e4 | Structured version Visualization version GIF version |
Description: Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 7508 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.) |
Ref | Expression |
---|---|
2p2e4 | ⊢ (2 + 2) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 10956 | . . 3 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq2i 6560 | . 2 ⊢ (2 + 2) = (2 + (1 + 1)) |
3 | df-4 10958 | . . 3 ⊢ 4 = (3 + 1) | |
4 | df-3 10957 | . . . 4 ⊢ 3 = (2 + 1) | |
5 | 4 | oveq1i 6559 | . . 3 ⊢ (3 + 1) = ((2 + 1) + 1) |
6 | 2cn 10968 | . . . 4 ⊢ 2 ∈ ℂ | |
7 | ax-1cn 9873 | . . . 4 ⊢ 1 ∈ ℂ | |
8 | 6, 7, 7 | addassi 9927 | . . 3 ⊢ ((2 + 1) + 1) = (2 + (1 + 1)) |
9 | 3, 5, 8 | 3eqtri 2636 | . 2 ⊢ 4 = (2 + (1 + 1)) |
10 | 2, 9 | eqtr4i 2635 | 1 ⊢ (2 + 2) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 (class class class)co 6549 1c1 9816 + caddc 9818 2c2 10947 3c3 10948 4c4 10949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-addass 9880 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-2 10956 df-3 10957 df-4 10958 |
This theorem is referenced by: 2t2e4 11054 i4 12829 4bc2eq6 12978 bpoly4 14629 fsumcube 14630 ef01bndlem 14753 6gcd4e2 15093 pythagtriplem1 15359 prmlem2 15665 43prm 15667 1259lem4 15679 2503lem1 15682 2503lem2 15683 2503lem3 15684 4001lem1 15686 4001lem4 15689 cphipval2 22848 quart1lem 24382 log2ub 24476 wallispi2lem1 38964 stirlinglem8 38974 sqwvfourb 39122 fmtnorec4 39999 m11nprm 40056 3exp4mod41 40071 gbogt5 40184 gbpart7 40189 sgoldbaltlem1 40201 sgoldbalt 40203 nnsum3primes4 40204 2t6m3t4e0 41919 2p2ne5 42353 |
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