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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremex-natded5.8 26401 Theorem 5.8 of [Clemente] 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 ((𝜓𝜒) → ¬ 𝜃) (𝜑 → ((𝜓𝜒) → ¬ 𝜃)) Given $e; adantr 479 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given $e; adantr 479 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given $e; adantr 479 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given $e. adantr 479 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 475. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 552 (I), 6,8 (adantr 479 to bring in scope)
75 ... ¬ 𝜃 ((𝜑𝜓) → ¬ 𝜃) E 1,6 mpd 15 (E), 2,4
812 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 ¬ 𝜓 (𝜑 → ¬ 𝜓) ¬I 5,7,8 pm2.65da 597 (¬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 𝜑 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 479; simpr 475 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 26402.

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

(𝜑 → ((𝜓𝜒) → ¬ 𝜃))    &   (𝜑 → (𝜏𝜃))    &   (𝜑𝜒)    &   (𝜑𝜏)       (𝜑 → ¬ 𝜓)
 
Theoremex-natded5.8-2 26402 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 26401. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → ¬ 𝜃))    &   (𝜑 → (𝜏𝜃))    &   (𝜑𝜒)    &   (𝜑𝜏)       (𝜑 → ¬ 𝜓)
 
Theoremex-natded5.13 26403 Theorem 5.13 of [Clemente] 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 26404. 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 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given $e.
2;32 (𝜓𝜃) (𝜑 → (𝜓𝜃)) Given $e. adantr 479 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given $e. ad2antrr 757 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 475
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 405 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 475
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 475
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 479 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 597 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 126 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 406 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 822 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 𝜑 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 479; simpr 475 is useful when you want to depend directly on the new assumption). (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (¬ 𝜏 → ¬ 𝜒))       (𝜑 → (𝜃𝜏))
 
Theoremex-natded5.13-2 26404 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 26403. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (¬ 𝜏 → ¬ 𝜒))       (𝜑 → (𝜃𝜏))
 
Theoremex-natded9.20 26405 Theorem 9.20 of [Clemente] 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 (𝜓 ∧ (𝜒𝜃)) (𝜑 → (𝜓 ∧ (𝜒𝜃))) Given $e
22 𝜓 (𝜑𝜓) EL 1 simpld 473 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 477 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 475
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 552 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 405 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 475
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 552 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 406 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 822 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 𝜑 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 479; simpr 475 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 26406. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 
Theoremex-natded9.20-2 26406 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 26405. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 
Theoremex-natded9.26 26407* Theorem 9.26 of [Clemente] 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 𝑥 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 𝑥𝑦𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝑦𝜓) Given $e.
26 ...| 𝑦𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) ND hypothesis assumption simpr 475. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3320 (E), 5,6. To use it we need a1i 11 and vex 3080. This could be immediately done with 19.21bi 2000, but we want to show the general approach for substitution.
412;8,9,10,11 ... 𝑥𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) I 3,a spesbcd 3392 (I), 11. To use it we need sylibr 222, which in turn requires sylib 206 and two uses of sbcid 3323. This could be more immediately done using 19.8a 1988, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2121 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1796 and nfe1 1969 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1808 (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 𝜑 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, 𝜓(𝑥, 𝑦) 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 26408.

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

(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)
 
Theoremex-natded9.26-2 26408* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 26407. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)
 
17.1.4  Definitional examples
 
Theoremex-or 26409 Example for df-or 383. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 3 ∨ 4 = 4)
 
Theoremex-an 26410 Example for df-an 384. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 2 ∧ 3 = 3)
 
Theoremex-dif 26411 Example for df-dif 3447. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∖ {1, 8}) = {3}
 
Theoremex-un 26412 Example for df-un 3449. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∪ {1, 8}) = {1, 3, 8}
 
Theoremex-in 26413 Example for df-in 3451. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∩ {1, 8}) = {1}
 
Theoremex-uni 26414 Example for df-uni 4271. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
{{1, 3}, {1, 8}} = {1, 3, 8}
 
Theoremex-ss 26415 Example for df-ss 3458. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊆ {1, 2, 3}
 
Theoremex-pss 26416 Example for df-pss 3460. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊊ {1, 2, 3}
 
Theoremex-pw 26417 Example for df-pw 4013. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
(𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))
 
Theoremex-pr 26418 Example for df-pr 4031. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐴 ∈ {1, -1} → (𝐴↑2) = 1)
 
Theoremex-br 26419 Example for df-br 4482. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)
 
Theoremex-opab 26420* Example for df-opab 4542. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)
 
Theoremex-eprel 26421 Example for df-eprel 4843. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
5 E {1, 5}
 
Theoremex-id 26422 Example for df-id 4847. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(5 I 5 ∧ ¬ 4 I 5)
 
Theoremex-po 26423 Example for df-po 4853. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
( < Po ℝ ∧ ¬ ≤ Po ℝ)
 
Theoremex-xp 26424 Example for df-xp 4938. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩})
 
Theoremex-cnv 26425 Example for df-cnv 4940. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}
 
Theoremex-co 26426 Example for df-co 4941. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((exp ∘ cos)‘0) = e
 
Theoremex-dm 26427 Example for df-dm 4942. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → dom 𝐹 = {2, 3})
 
Theoremex-rn 26428 Example for df-rn 4943. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})
 
Theoremex-res 26429 Example for df-res 4944. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {⟨2, 6⟩})
 
Theoremex-ima 26430 Example for df-ima 4945. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {6})
 
Theoremex-fv 26431 Example for df-fv 5697. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9)
 
Theoremex-1st 26432 Example for df-1st 6933. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(1st ‘⟨3, 4⟩) = 3
 
Theoremex-2nd 26433 Example for df-2nd 6934. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(2nd ‘⟨3, 4⟩) = 4
 
Theorem1kp2ke3k 26434 Example for df-dec 11233, 1000 + 2000 = 3000.

This proof disproves (by counterexample) 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 (2 + 1) = 3, 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 11233 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.)

(1000 + 2000) = 3000
 
Theoremex-fl 26435 Example for df-fl 12322. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)
 
Theoremex-ceil 26436 Example for df-ceil 12323. (Contributed by AV, 4-Sep-2021.)
((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)
 
Theoremex-mod 26437 Example for df-mod 12398. (Contributed by AV, 3-Sep-2021.)
((5 mod 3) = 2 ∧ (-7 mod 2) = 1)
 
Theoremex-exp 26438 Example for df-exp 12590. (Contributed by AV, 4-Sep-2021.)
((5↑2) = 25 ∧ (-3↑-2) = (1 / 9))
 
Theoremex-fac 26439 Example for df-fac 12790. (Contributed by AV, 4-Sep-2021.)
(!‘5) = 120
 
Theoremex-bc 26440 Example for df-bc 12819. (Contributed by AV, 4-Sep-2021.)
(5C3) = 10
 
Theoremex-hash 26441 Example for df-hash 12847. (Contributed by AV, 4-Sep-2021.)
(#‘{0, 1, 2}) = 3
 
Theoremex-sqrt 26442 Example for df-sqrt 13680. (Contributed by AV, 4-Sep-2021.)
(√‘25) = 5
 
Theoremex-abs 26443 Example for df-abs 13681. (Contributed by AV, 4-Sep-2021.)
(abs‘-2) = 2
 
Theoremex-dvds 26444 Example for df-dvds 14689: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
3 ∥ 6
 
Theoremex-gcd 26445 Example for df-gcd 14926. (Contributed by AV, 5-Sep-2021.)
(-6 gcd 9) = 3
 
Theoremex-lcm 26446 Example for df-lcm 15015. (Contributed by AV, 5-Sep-2021.)
(6 lcm 9) = 18
 
Theoremex-prmo 26447 Example for df-prmo 15456: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.)
(#p10) = 210
 
17.1.5  Other examples
 
Theoremaevdemo 26448* Proof illustrating the comment of aev2 1934. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
 
Theoremex-ind-dvds 26449 Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
(𝑁 ∈ ℕ0 → 3 ∥ ((4↑𝑁) + 2))
 
17.2  Humor
 
17.2.1  April Fool's theorem
 
Theoremavril1 26450 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object 𝑥 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 mmnotes.txt, under the 1-Apr-2006 entry.

¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))
 
Theorem2bornot2b 26451 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.)
(2 · 𝐵 ∨ ¬ 2 · 𝐵)
 
Theoremhelloworld 26452 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.)
¬ ( ∈ (𝐿𝐿0) ∧ 𝑊∅(R1𝑑))
 
Theorem1p1e2apr1 26453 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel L. 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.)
(1 + 1) = 2
 
Theoremeqid1 26454 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.)

𝐴 = 𝐴
 
Theorem1div0apr 26455 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.)
(1 / 0) = ∅
 
Theoremtopnfbey 26456 Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Modified by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ (0...+∞) → +∞ < 𝐵)
 
17.3  (Future - to be reviewed and classified)
 
17.3.1  Planar incidence geometry
 
Syntaxcplig 26457 Extend class notation with the class of all planar incidence geometries.
class Plig
 
Definitiondf-plig 26458* Define the class of planar incidence geometries. We use Hilbert's axioms and adapt them to planar geometry. We use for the incidence relation. We could have used a generic binary relation, but using allows us to reuse previous results. Much of what follows is directly borrowed from Aitken, Incidence-Betweenness Geometry, 2008, http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf.

The class Plig is the class of planar incidence geometries, where a planar incidence geometry 𝑥 is defined as a set of lines 𝑙 satisfying three axioms. In the definition below, 𝑥 is the union of lines, that is, the plane, and 𝑎, 𝑏, 𝑐 denote points. Therefore, the axioms are: for all pairs of (distinct) points, there exists a unique line containing them; all lines contain at least two points; there exist three non-collinear points.

(Contributed by FL, 2-Aug-2009.)

Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
 
Theoremisplig 26459* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
𝑃 = 𝐿       (𝐿𝐴 → (𝐿 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐿 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐿𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
 
Theoremtncp 26460* In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.)
𝑃 = 𝐿       (𝐿 ∈ Plig → ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐿 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))
 
Theoremlpni 26461* For any line in a planar incidence geometry, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
𝑃 = 𝐺       ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃 𝑎𝐿)
 
17.3.2  Algebra preliminaries
 
Syntaxcrpm 26462 Ring primes.
class RPrime
 
Definitiondf-rprm 26463* 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 15137. (Contributed by Mario Carneiro, 17-Feb-2015.)
RPrime = (𝑤 ∈ V ↦ (Base‘𝑤) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g𝑤)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑤) / 𝑑](𝑝𝑑(𝑥(.r𝑤)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))})
 
17.3.3  Aliases kept to prevent broken links

A few aliases that we temporarily keep to prevent broken links. If you land on any of these, please let the originating site and/or us know that the link that made you land here should be changed.

 
Theoremdummylink 26464 Alias for a1ii 1 that may be referenced in some older works, and kept here to prevent broken links.

If you landed here, please let the originating site and/or us know that the link that made you land here should be changed to a link to a1ii 1.

(Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑    &   𝜓       𝜑
 
Theoremid1 26465 Alias for idALT 23 that may be referenced in some older works, and kept here to prevent broken links.

If you landed here, please let the originating site and/or us know that the link that made you land here should be changed to a link to idALT 23.

(Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑𝜑)
 
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)

The intent is for this deprecated section to be deleted once its theorems have extensible structure versions (or are not useful). You can make a list of "terminal" theorems (i.e. theorems not referenced by anything else) and for each theorem see if there exists an extensible structure version (or decide it's not useful), and if so, delete it. Then repeat this recursively. One way to search for terminal theorems is to log the output ("open log x.txt") of "show usage WHATEVER" in metamath.exe and search for "(None)".

 
18.1  Additional material on group theory (deprecated)

This section contains an earlier development of groups that was defined before extensible structures were introduced.

The intent is for this deprecated section to be deleted once the corresponding definitions and theorems for complex topological vector spaces, which are using them, are revised accordingly.

 
18.1.1  Definitions and basic properties for groups
 
Syntaxcgr 26466 Extend class notation with the class of all group operations.
class GrpOp
 
Syntaxcgi 26467 Extend class notation with a function mapping a group operation to the group's identity element.
class GId
 
Syntaxcgn 26468 Extend class notation with a function mapping a group operation to the inverse function for the group.
class inv
 
Syntaxcgs 26469 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
class /𝑔
 
Definitiondf-grpo 26470* 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.)
GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
 
Definitiondf-gid 26471* 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 = (𝑔 ∈ V ↦ (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
 
Definitiondf-ginv 26472* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
 
Definitiondf-gdiv 26473* 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.)
/𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
 
Theoremisgrpo 26474* The predicate "is a group operation." Note that 𝑋 is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺𝐴 → (𝐺 ∈ GrpOp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))))
 
Theoremisgrpoi 26475* Properties that determine a group operation. Read 𝑁 as 𝑁(𝑥). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝑋 ∈ V    &   𝐺:(𝑋 × 𝑋)⟶𝑋    &   ((𝑥𝑋𝑦𝑋𝑧𝑋) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   𝑈𝑋    &   (𝑥𝑋 → (𝑈𝐺𝑥) = 𝑥)    &   (𝑥𝑋𝑁𝑋)    &   (𝑥𝑋 → (𝑁𝐺𝑥) = 𝑈)       𝐺 ∈ GrpOp
 
Theoremgrpofo 26476 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → 𝐺:(𝑋 × 𝑋)–onto𝑋)
 
Theoremgrpocl 26477 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
 
Theoremgrpolidinv 26478* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ ∃𝑦𝑋 (𝑦𝐺𝑥) = 𝑢))
 
Theoremgrpon0 26479 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → 𝑋 ≠ ∅)
 
Theoremgrpoass 26480 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
 
Theoremgrpoidinvlem1 26481 Lemma for grpoidinv 26485. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑌𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝐴) = 𝐴)) → (𝑈𝐺𝐴) = 𝑈)
 
Theoremgrpoidinvlem2 26482 Lemma for grpoidinv 26485. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (((𝐺 ∈ GrpOp ∧ (𝑌𝑋𝐴𝑋)) ∧ ((𝑈𝐺𝑌) = 𝑌 ∧ (𝑌𝐺𝐴) = 𝑈)) → ((𝐴𝐺𝑌)𝐺(𝐴𝐺𝑌)) = (𝐴𝐺𝑌))
 
Theoremgrpoidinvlem3 26483* Lemma for grpoidinv 26485. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   (𝜑 ↔ ∀𝑥𝑋 (𝑈𝐺𝑥) = 𝑥)    &   (𝜓 ↔ ∀𝑥𝑋𝑧𝑋 (𝑧𝐺𝑥) = 𝑈)       ((((𝐺 ∈ GrpOp ∧ 𝑈𝑋) ∧ (𝜑𝜓)) ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
 
Theoremgrpoidinvlem4 26484* Lemma for grpoidinv 26485. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → (𝐴𝐺𝑈) = (𝑈𝐺𝐴))
 
Theoremgrpoidinv 26485* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → ∃𝑢𝑋𝑥𝑋 (((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥) ∧ ∃𝑦𝑋 ((𝑦𝐺𝑥) = 𝑢 ∧ (𝑥𝐺𝑦) = 𝑢)))
 
Theoremgrpoideu 26486* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ GrpOp → ∃!𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥)
 
Theoremgrporndm 26487 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
(𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)
 
Theorem0ngrp 26488 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
¬ ∅ ∈ GrpOp
 
Theoremgidval 26489* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺𝑉 → (GId‘𝐺) = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
 
Theoremgrpoidval 26490* Lemma for grpoidcl 26491 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       (𝐺 ∈ GrpOp → 𝑈 = (𝑢𝑋𝑥𝑋 (𝑢𝐺𝑥) = 𝑥))
 
Theoremgrpoidcl 26491 The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       (𝐺 ∈ GrpOp → 𝑈𝑋)
 
Theoremgrpoidinv2 26492* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
 
Theoremgrpolid 26493 The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑈𝐺𝐴) = 𝐴)
 
Theoremgrporid 26494 The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑈) = 𝐴)
 
Theoremgrporcan 26495 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
 
Theoremgrpoinveu 26496* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
 
Theoremgrpoid 26497 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))
 
Theoremgrporn 26498 The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
𝐺 ∈ GrpOp    &   dom 𝐺 = (𝑋 × 𝑋)       𝑋 = ran 𝐺
 
Theoremgrpoinvfval 26499* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
 
Theoremgrpoinvval 26500* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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