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Type | Label | Description | ||||||||||||
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Statement | ||||||||||||||
Theorem | redivapd 9201 | Real part of a division. Related to remul2 9101. (Contributed by Jim Kingdon, 15-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (ℜ‘(B / A)) = ((ℜ‘B) / A)) | ||||||||||||||
Theorem | imdivapd 9202 | Imaginary part of a division. Related to remul2 9101. (Contributed by Jim Kingdon, 15-Jun-2020.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℂ) & ⊢ (φ → A # 0) ⇒ ⊢ (φ → (ℑ‘(B / A)) = ((ℑ‘B) / A)) | ||||||||||||||
Theorem | crred 9203 | The real part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℝ) ⇒ ⊢ (φ → (ℜ‘(A + (i · B))) = A) | ||||||||||||||
Theorem | crimd 9204 | The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ (φ → A ∈ ℝ) & ⊢ (φ → B ∈ ℝ) ⇒ ⊢ (φ → (ℑ‘(A + (i · B))) = B) | ||||||||||||||
Syntax | csqrt 9205 | Extend class notation to include square root of a complex number. | ||||||||||||
class √ | ||||||||||||||
Syntax | cabs 9206 | Extend class notation to include a function for the absolute value (modulus) of a complex number. | ||||||||||||
class abs | ||||||||||||||
Definition | df-rsqrt 9207* |
Define a function whose value is the square root of a nonnegative real
number.
Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root. (Contributed by Jim Kingdon, 23-Aug-2020.) | ||||||||||||
⊢ √ = (x ∈ ℝ ↦ (℩y ∈ ℝ ((y↑2) = x ∧ 0 ≤ y))) | ||||||||||||||
Definition | df-abs 9208 | Define the function for the absolute value (modulus) of a complex number. (Contributed by NM, 27-Jul-1999.) | ||||||||||||
⊢ abs = (x ∈ ℂ ↦ (√‘(x · (∗‘x)))) | ||||||||||||||
Theorem | sqrtrval 9209* | Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.) | ||||||||||||
⊢ (A ∈ ℝ → (√‘A) = (℩x ∈ ℝ ((x↑2) = A ∧ 0 ≤ x))) | ||||||||||||||
Theorem | absval 9210 | The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of [Gleason] p. 133. (Contributed by NM, 27-Jul-1999.) (Revised by Mario Carneiro, 7-Nov-2013.) | ||||||||||||
⊢ (A ∈ ℂ → (abs‘A) = (√‘(A · (∗‘A)))) | ||||||||||||||
Theorem | rennim 9211 | A real number does not lie on the negative imaginary axis. (Contributed by Mario Carneiro, 8-Jul-2013.) | ||||||||||||
⊢ (A ∈ ℝ → (i · A) ∉ ℝ^{+}) | ||||||||||||||
Theorem | sqrt0rlem 9212 | Lemma for sqrt0 9213. (Contributed by Jim Kingdon, 26-Aug-2020.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ ((A↑2) = 0 ∧ 0 ≤ A)) ↔ A = 0) | ||||||||||||||
Theorem | sqrt0 9213 | Square root of zero. (Contributed by Mario Carneiro, 9-Jul-2013.) | ||||||||||||
⊢ (√‘0) = 0 | ||||||||||||||
Theorem | sqrtsq 9214 | Square root of square. (Contributed by NM, 14-Jan-2006.) (Revised by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (√‘(A↑2)) = A) | ||||||||||||||
Theorem | sqrtmsq 9215 | Square root of square. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (√‘(A · A)) = A) | ||||||||||||||
Theorem | sqrt1 9216 | The square root of 1 is 1. (Contributed by NM, 31-Jul-1999.) | ||||||||||||
⊢ (√‘1) = 1 | ||||||||||||||
Theorem | sqrt4 9217 | The square root of 4 is 2. (Contributed by NM, 3-Aug-1999.) | ||||||||||||
⊢ (√‘4) = 2 | ||||||||||||||
Theorem | sqrt9 9218 | The square root of 9 is 3. (Contributed by NM, 11-May-2004.) | ||||||||||||
⊢ (√‘9) = 3 | ||||||||||||||
Theorem | absneg 9219 | Absolute value of negative. (Contributed by NM, 27-Feb-2005.) | ||||||||||||
⊢ (A ∈ ℂ → (abs‘-A) = (abs‘A)) | ||||||||||||||
Theorem | abscj 9220 | The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by NM, 28-Apr-2005.) | ||||||||||||
⊢ (A ∈ ℂ → (abs‘(∗‘A)) = (abs‘A)) | ||||||||||||||
Theorem | absval2 9221 | Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by NM, 17-Mar-2005.) | ||||||||||||
⊢ (A ∈ ℂ → (abs‘A) = (√‘(((ℜ‘A)↑2) + ((ℑ‘A)↑2)))) | ||||||||||||||
Theorem | abs0 9222 | The absolute value of 0. (Contributed by NM, 26-Mar-2005.) (Revised by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ (abs‘0) = 0 | ||||||||||||||
Theorem | absi 9223 | The absolute value of the imaginary unit. (Contributed by NM, 26-Mar-2005.) | ||||||||||||
⊢ (abs‘i) = 1 | ||||||||||||||
Theorem | abs00bd 9224 | If a complex number is zero, its absolute value is zero. (Contributed by David Moews, 28-Feb-2017.) | ||||||||||||
⊢ (φ → A = 0) ⇒ ⊢ (φ → (abs‘A) = 0) | ||||||||||||||
Theorem | absid 9225 | A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ 0 ≤ A) → (abs‘A) = A) | ||||||||||||||
Theorem | abs1 9226 | The absolute value of 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) | ||||||||||||
⊢ (abs‘1) = 1 | ||||||||||||||
Theorem | absnid 9227 | A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005.) | ||||||||||||
⊢ ((A ∈ ℝ ∧ A ≤ 0) → (abs‘A) = -A) | ||||||||||||||
Theorem | absre 9228 | Absolute value of a real number. (Contributed by NM, 17-Mar-2005.) | ||||||||||||
⊢ (A ∈ ℝ → (abs‘A) = (√‘(A↑2))) | ||||||||||||||
Theorem | nn0abscl 9229 | The absolute value of an integer is a nonnegative integer. (Contributed by NM, 27-Feb-2005.) | ||||||||||||
⊢ (A ∈ ℤ → (abs‘A) ∈ ℕ_{0}) | ||||||||||||||
Theorem | zabscl 9230 | The absolute value of an integer is an integer. (Contributed by Stefan O'Rear, 24-Sep-2014.) | ||||||||||||
⊢ (A ∈ ℤ → (abs‘A) ∈ ℤ) | ||||||||||||||
This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:
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Theorem | conventions 9231 |
Unless there is a reason to diverge, we follow the conventions of
the Metamath Proof Explorer (aka "set.mm"). This list of
conventions is intended to be read in conjunction with the
corresponding conventions in the Metamath Proof Explorer, and
only the differences are described below.
Label naming conventions Here are a few of the label naming conventions:
The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.
(Contributed by Jim Kingdon, 24-Feb-2020.) | ||||||||||||
⊢ φ ⇒ ⊢ φ | ||||||||||||||
Theorem | mathbox 9232 |
(This theorem is a dummy placeholder for these guidelines. The name of
this theorem, "mathbox", is hard-coded into the Metamath program
to
identify the start of the mathbox section for web page generation.)
A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm. Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it. Guidelines: 1. If at all possible, please use only 0-ary class constants for new definitions. 2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website. 3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm. 4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed. Notes: 1. We may decide to move some theorems to the main part of set.mm for general use. 2. We may make changes to mathboxes to maintain the overall quality of set.mm. Normally we will let you know if a change might impact what you are working on. 3. If you use theorems from another user's mathbox, we don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let us know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.) | ||||||||||||
⊢ x = x | ||||||||||||||
Theorem | ax1hfs 9233 | Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.) | ||||||||||||
⊢ (φ → (φ ∧ φ)) | ||||||||||||||
Theorem | nnexmid 9234 | Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.) | ||||||||||||
⊢ ¬ ¬ (φ ∨ ¬ φ) | ||||||||||||||
Theorem | nndc 9235 | Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.) | ||||||||||||
⊢ ¬ ¬ DECID φ | ||||||||||||||
Theorem | dcdc 9236 | Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.) | ||||||||||||
⊢ (DECID DECID φ ↔ DECID φ) | ||||||||||||||
Theorem | bj-ex 9237* | Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1486 and 19.9ht 1529 or 19.23ht 1383). (Proof modification is discouraged.) | ||||||||||||
⊢ (∃xφ → φ) | ||||||||||||||
Theorem | bj-hbalt 9238 | Closed form of hbal 1363 (copied from set.mm). (Contributed by BJ, 2-May-2019.) | ||||||||||||
⊢ (∀y(φ → ∀xφ) → (∀yφ → ∀x∀yφ)) | ||||||||||||||
Theorem | bj-nfalt 9239 | Closed form of nfal 1465 (copied from set.mm). (Contributed by BJ, 2-May-2019.) | ||||||||||||
⊢ (∀xℲyφ → Ⅎy∀xφ) | ||||||||||||||
Theorem | spimd 9240 | Deduction form of spim 1623. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ (φ → Ⅎxχ) & ⊢ (φ → ∀x(x = y → (ψ → χ))) ⇒ ⊢ (φ → (∀xψ → χ)) | ||||||||||||||
Theorem | 2spim 9241* | Double substitution, as in spim 1623. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ Ⅎxχ & ⊢ Ⅎzχ & ⊢ ((x = y ∧ z = 𝑡) → (ψ → χ)) ⇒ ⊢ (∀z∀xψ → χ) | ||||||||||||||
Theorem | ch2var 9242* | Implicit substitution of y for x and 𝑡 for z into a theorem. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ Ⅎzψ & ⊢ ((x = y ∧ z = 𝑡) → (φ ↔ ψ)) & ⊢ φ ⇒ ⊢ ψ | ||||||||||||||
Theorem | ch2varv 9243* | Version of ch2var 9242 with non-freeness hypotheses replaced by DV conditions. (Contributed by BJ, 17-Oct-2019.) | ||||||||||||
⊢ ((x = y ∧ z = 𝑡) → (φ ↔ ψ)) & ⊢ φ ⇒ ⊢ ψ | ||||||||||||||
Theorem | bj-exlimmp 9244 | Lemma for bj-vtoclgf 9250. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (χ → φ) ⇒ ⊢ (∀x(χ → (φ → ψ)) → (∃xχ → ψ)) | ||||||||||||||
Theorem | bj-exlimmpi 9245 | Lemma for bj-vtoclgf 9250. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (χ → φ) & ⊢ (χ → (φ → ψ)) ⇒ ⊢ (∃xχ → ψ) | ||||||||||||||
Theorem | bj-sbimedh 9246 | A strengthening of sbiedh 1667 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ (φ → ∀xφ) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (x = y → (ψ → χ))) ⇒ ⊢ (φ → ([y / x]ψ → χ)) | ||||||||||||||
Theorem | bj-sbimeh 9247 | A strengthening of sbieh 1670 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ ([y / x]φ → ψ) | ||||||||||||||
Theorem | bj-sbime 9248 | A strengthening of sbie 1671 (same proof). (Contributed by BJ, 16-Dec-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ ([y / x]φ → ψ) | ||||||||||||||
Various utility theorems using FOL and extensionality. | ||||||||||||||
Theorem | bj-vtoclgft 9249 | Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → φ) ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ 𝑉 → ψ)) | ||||||||||||||
Theorem | bj-vtoclgf 9250 | Weakening two hypotheses of vtoclgf 2606. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → φ) & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ 𝑉 → ψ) | ||||||||||||||
Theorem | elabgf0 9251 | Lemma for elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (A ∈ {x ∣ φ} ↔ φ)) | ||||||||||||||
Theorem | elabgft1 9252 | One implication of elabgf 2679, in closed form. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ {x ∣ φ} → ψ)) | ||||||||||||||
Theorem | elabgf1 9253 | One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
Theorem | elabgf2 9254 | One implication of elabgf 2679. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ B → (ψ → A ∈ {x ∣ φ})) | ||||||||||||||
Theorem | elabf1 9255* | One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
Theorem | elabf2 9256* | One implication of elabf 2680. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ Ⅎxψ & ⊢ A ∈ V & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (ψ → A ∈ {x ∣ φ}) | ||||||||||||||
Theorem | elab1 9257* | One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (φ → ψ)) ⇒ ⊢ (A ∈ {x ∣ φ} → ψ) | ||||||||||||||
Theorem | elab2a 9258* | One implication of elab 2681. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ A ∈ V & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (ψ → A ∈ {x ∣ φ}) | ||||||||||||||
Theorem | elabg2 9259* | One implication of elabg 2682. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ 𝑉 → (ψ → A ∈ {x ∣ φ})) | ||||||||||||||
Theorem | bj-rspgt 9260 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ ℲxB & ⊢ Ⅎxψ ⇒ ⊢ (∀x(x = A → (φ → ψ)) → (∀x ∈ B φ → (A ∈ B → ψ))) | ||||||||||||||
Theorem | bj-rspg 9261 | Restricted specialization, generalized. Weakens a hypothesis of rspccv 2647 and seems to have a shorter proof. (Contributed by BJ, 21-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ ℲxB & ⊢ Ⅎxψ & ⊢ (x = A → (φ → ψ)) ⇒ ⊢ (∀x ∈ B φ → (A ∈ B → ψ)) | ||||||||||||||
Theorem | cbvrald 9262* | Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.) | ||||||||||||
⊢ Ⅎxφ & ⊢ Ⅎyφ & ⊢ (φ → Ⅎyψ) & ⊢ (φ → Ⅎxχ) & ⊢ (φ → (x = y → (ψ ↔ χ))) ⇒ ⊢ (φ → (∀x ∈ A ψ ↔ ∀y ∈ A χ)) | ||||||||||||||
Theorem | bj-intabssel 9263 | Version of intss1 3621 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA ⇒ ⊢ (A ∈ 𝑉 → ([A / x]φ → ∩ {x ∣ φ} ⊆ A)) | ||||||||||||||
Theorem | bj-intabssel1 9264 | Version of intss1 3621 using a class abstraction and implicit substitution. Closed form of intmin3 3633. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA & ⊢ Ⅎxψ & ⊢ (x = A → (ψ → φ)) ⇒ ⊢ (A ∈ 𝑉 → (ψ → ∩ {x ∣ φ} ⊆ A)) | ||||||||||||||
Theorem | bj-elssuniab 9265 | Version of elssuni 3599 using a class abstraction and explicit substitution. (Contributed by BJ, 29-Nov-2019.) | ||||||||||||
⊢ ℲxA ⇒ ⊢ (A ∈ 𝑉 → ([A / x]φ → A ⊆ ∪ {x ∣ φ})) | ||||||||||||||
Theorem | bj-sseq 9266 | If two converse inclusions are characterized each by a formula, then equality is characterized by the conjunction of these formulas. (Contributed by BJ, 30-Nov-2019.) | ||||||||||||
⊢ (φ → (ψ ↔ A ⊆ B)) & ⊢ (φ → (χ ↔ B ⊆ A)) ⇒ ⊢ (φ → ((ψ ∧ χ) ↔ A = B)) | ||||||||||||||
This is an ongoing project to define bounded formulas, following a discussion on GitHub between Jim Kingdon, Mario Carneiro and I, started 23-Sept-2019 (see https://github.com/metamath/set.mm/issues/1173 and links therein). In order to state certain axiom schemes of Constructive Zermelo–Fraenkel (CZF) set theory, like the axiom scheme of bounded (or restricted, or Δ_{0}) separation, it is necessary to distinguish certain formulas, called bounded (or restricted, or Δ_{0}) formulas. The necessity of considering bounded formulas also arises in several theories of bounded arithmetic, both classical or intuitonistic, for instance to state the axiom scheme of Δ_{0}-induction. To formalize this in Metamath, there are several choices to make. A first choice is to either create a new type for bounded formulas, or to create a predicate on formulas that indicates whether they are bounded. In the first case, one creates a new type "wff0" with a new set of metavariables (ph_{0} ...) and an axiom "$a wff ph_{0} " ensuring that bounded formulas are formulas, so that one can reuse existing theorems, and then axioms take the form "$a wff0 ( ph_{0} -> ps_{0} )", etc. In the second case, one introduces a predicate "BOUNDED " with the intended meaning that "BOUNDED φ " is a formula meaning that φ is a bounded formula. We choose the second option, since the first would complicate the grammar, risking to make it ambiguous. (TODO: elaborate.) A second choice is to view "bounded" either as a syntactic or a semantic property. For instance, ∀x ⊤ is not syntactically bounded since it has an unbounded universal quantifier, but it is semantically bounded since it is equivalent to ⊤ which is bounded. We choose the second option, so that formulas using defined symbols can be proved bounded. A third choice is in the form of the axioms, either in closed form or in inference form. One cannot state all the axioms in closed form, especially ax-bd0 9268. Indeed, if we posited it in closed form, then we could prove for instance ⊢ (φ → BOUNDED φ) and ⊢ (¬ φ → BOUNDED φ) which is problematic (with the law of excluded middle, this would entail that all formulas are bounded, but even without it, too many formulas could be proved bounded...). (TODO: elaborate.) Having ax-bd0 9268 in inference form ensures that a formula can be proved bounded only if it is equivalent *for all values of the free variables* to a syntactically bounded one. The other axioms (ax-bdim 9269 through ax-bdsb 9277) can be written either in closed or inference form. The fact that ax-bd0 9268 is an inference is enough to ensure that the closed forms cannot be "exploited" to prove that some unbounded formulas are bounded. (TODO: check.) However, we state all the axioms in inference form to make it clear that we do not exploit any over-permissiveness. Finally, note that our logic has no terms, only variables. Therefore, we cannot prove for instance that x ∈ 𝜔 is a bounded formula. However, since 𝜔 can be defined as "the y such that PHI" a proof using the fact that x ∈ 𝜔 is bounded can be converted to a proof in iset.mm by replacing 𝜔 with y everywhere and prepending the antecedent PHI, since x ∈ y is bounded by ax-bdel 9276. For a similar method, see bj-omtrans 9416. Note that one cannot add an axiom ⊢ BOUNDED x ∈ A since by bdph 9305 it would imply that every formula is bounded. For CZF, a useful set of notes is Peter Aczel and Michael Rathjen, CST Book draft. (available at http://www1.maths.leeds.ac.uk/~rathjen/book.pdf) and an interesting article is Michael Shulman, Comparing material and structural set theories, Annals of Pure and Applied Logic, Volume 170, Issue 4 (Apr. 2019), 465--504. (available at https://arxiv.org/abs/1808.05204) | ||||||||||||||
Syntax | wbd 9267 | Syntax for the predicate BOUNDED. | ||||||||||||
wff BOUNDED φ | ||||||||||||||
Axiom | ax-bd0 9268 | If two formulas are equivalent, then boundedness of one implies boundedness of the other. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ (φ ↔ ψ) ⇒ ⊢ (BOUNDED φ → BOUNDED ψ) | ||||||||||||||
Axiom | ax-bdim 9269 | An implication between two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ → ψ) | ||||||||||||||
Axiom | ax-bdan 9270 | The conjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ∧ ψ) | ||||||||||||||
Axiom | ax-bdor 9271 | The disjunction of two bounded formulas is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ∨ ψ) | ||||||||||||||
Axiom | ax-bdn 9272 | The negation of a bounded formula is bounded. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED ¬ φ | ||||||||||||||
Axiom | ax-bdal 9273* | A bounded universal quantification of a bounded formula is bounded. Note the DV condition on x, y. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED ∀x ∈ y φ | ||||||||||||||
Axiom | ax-bdex 9274* | A bounded existential quantification of a bounded formula is bounded. Note the DV condition on x, y. (Contributed by BJ, 25-Sep-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED ∃x ∈ y φ | ||||||||||||||
Axiom | ax-bdeq 9275 | An atomic formula is bounded (equality predicate). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED x = y | ||||||||||||||
Axiom | ax-bdel 9276 | An atomic formula is bounded (membership predicate). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED x ∈ y | ||||||||||||||
Axiom | ax-bdsb 9277 | A formula resulting from proper substitution in a bounded formula is bounded. This probably cannot be proved from the other axioms, since neither the definiens in df-sb 1643, nor probably any other equivalent formula, is syntactically bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED [y / x]φ | ||||||||||||||
Theorem | bdeq 9278 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ (φ ↔ ψ) ⇒ ⊢ (BOUNDED φ ↔ BOUNDED ψ) | ||||||||||||||
Theorem | bd0 9279 | A formula equivalent to a bounded one is bounded. See also bd0r 9280. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ (φ ↔ ψ) ⇒ ⊢ BOUNDED ψ | ||||||||||||||
Theorem | bd0r 9280 | A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 9279) biconditional in the hypothesis, to work better with definitions (ψ is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ (ψ ↔ φ) ⇒ ⊢ BOUNDED ψ | ||||||||||||||
Theorem | bdbi 9281 | A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ↔ ψ) | ||||||||||||||
Theorem | bdstab 9282 | Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED STAB φ | ||||||||||||||
Theorem | bddc 9283 | Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED DECID φ | ||||||||||||||
Theorem | bd3or 9284 | A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ & ⊢ BOUNDED χ ⇒ ⊢ BOUNDED (φ ∨ ψ ∨ χ) | ||||||||||||||
Theorem | bd3an 9285 | A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ & ⊢ BOUNDED χ ⇒ ⊢ BOUNDED (φ ∧ ψ ∧ χ) | ||||||||||||||
Theorem | bdth 9286 | A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.) | ||||||||||||
⊢ φ ⇒ ⊢ BOUNDED φ | ||||||||||||||
Theorem | bdtru 9287 | The truth value ⊤ is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED ⊤ | ||||||||||||||
Theorem | bdfal 9288 | The truth value ⊥ is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED ⊥ | ||||||||||||||
Theorem | bdnth 9289 | A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.) | ||||||||||||
⊢ ¬ φ ⇒ ⊢ BOUNDED φ | ||||||||||||||
Theorem | bdnthALT 9290 | Alternate proof of bdnth 9289 not using bdfal 9288. Then, bdfal 9288 can be proved from this theorem, using fal 1249. The total number of proof steps would be 17 (for bdnthALT 9290) + 3 = 20, which is more than 8 (for bdfal 9288) + 9 (for bdnth 9289) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | ||||||||||||
⊢ ¬ φ ⇒ ⊢ BOUNDED φ | ||||||||||||||
Theorem | bdxor 9291 | The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ & ⊢ BOUNDED ψ ⇒ ⊢ BOUNDED (φ ⊻ ψ) | ||||||||||||||
Theorem | bj-bdcel 9292* | Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.) | ||||||||||||
⊢ BOUNDED y = A ⇒ ⊢ BOUNDED A ∈ x | ||||||||||||||
Theorem | bdab 9293 | Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED x ∈ {y ∣ φ} | ||||||||||||||
Theorem | bdcdeq 9294 | Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) | ||||||||||||
⊢ BOUNDED φ ⇒ ⊢ BOUNDED CondEq(x = y → φ) | ||||||||||||||
In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 9296. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas. As will be clear by the end of this subsection (see for instance bdop 9330), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance, ⊢ BOUNDED φ ⇒ ⊢ BOUNDED ⟨{x ∣ φ}, ({y, suc z} × ⟨𝑡, ∅⟩)⟩. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like ⊢ BOUNDED A ⇒ ⊢ BOUNDED {A}. | ||||||||||||||
Syntax | wbdc 9295 | Syntax for the predicate BOUNDED. | ||||||||||||
wff BOUNDED A | ||||||||||||||
Definition | df-bdc 9296* | Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ (BOUNDED A ↔ ∀xBOUNDED x ∈ A) | ||||||||||||||
Theorem | bdceq 9297 | Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ A = B ⇒ ⊢ (BOUNDED A ↔ BOUNDED B) | ||||||||||||||
Theorem | bdceqi 9298 | A class equal to a bounded one is bounded. Note the use of ax-ext 2019. See also bdceqir 9299. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ A = B ⇒ ⊢ BOUNDED B | ||||||||||||||
Theorem | bdceqir 9299 | A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 9298) equality in the hypothesis, to work better with definitions (B is the definiendum that one wants to prove bounded; see comment of bd0r 9280). (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ BOUNDED A & ⊢ B = A ⇒ ⊢ BOUNDED B | ||||||||||||||
Theorem | bdel 9300* | The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.) | ||||||||||||
⊢ (BOUNDED A → BOUNDED x ∈ A) |
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