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Theorem caovmo 7366
 Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)
Hypotheses
Ref Expression
caovmo.2 𝐵𝑆
caovmo.dom dom 𝐹 = (𝑆 × 𝑆)
caovmo.3 ¬ ∅ ∈ 𝑆
caovmo.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caovmo.ass ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
caovmo.id (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
Assertion
Ref Expression
caovmo ∃*𝑤(𝐴𝐹𝑤) = 𝐵
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦   𝑤,𝐵,𝑧   𝑤,𝐹   𝑤,𝑆

Proof of Theorem caovmo
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7142 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝐹𝑤) = (𝐴𝐹𝑤))
21eqeq1d 2800 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑤) = 𝐵))
32mobidv 2608 . . . 4 (𝑢 = 𝐴 → (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
4 oveq2 7143 . . . . . . 7 (𝑤 = 𝑣 → (𝑢𝐹𝑤) = (𝑢𝐹𝑣))
54eqeq1d 2800 . . . . . 6 (𝑤 = 𝑣 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝑢𝐹𝑣) = 𝐵))
65mo4 2625 . . . . 5 (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∀𝑤𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣))
7 simpr 488 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) = 𝐵)
87oveq2d 7151 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑤𝐹𝐵))
9 simpl 486 . . . . . . . . . 10 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) = 𝐵)
109oveq1d 7150 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → ((𝑢𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
11 vex 3444 . . . . . . . . . . 11 𝑢 ∈ V
12 vex 3444 . . . . . . . . . . 11 𝑤 ∈ V
13 vex 3444 . . . . . . . . . . 11 𝑣 ∈ V
14 caovmo.ass . . . . . . . . . . 11 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
1511, 12, 13, 14caovass 7329 . . . . . . . . . 10 ((𝑢𝐹𝑤)𝐹𝑣) = (𝑢𝐹(𝑤𝐹𝑣))
16 caovmo.com . . . . . . . . . . 11 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
1711, 12, 13, 16, 14caov12 7357 . . . . . . . . . 10 (𝑢𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝑢𝐹𝑣))
1815, 17eqtri 2821 . . . . . . . . 9 ((𝑢𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝑢𝐹𝑣))
19 caovmo.2 . . . . . . . . . . 11 𝐵𝑆
2019elexi 3460 . . . . . . . . . 10 𝐵 ∈ V
2120, 13, 16caovcom 7326 . . . . . . . . 9 (𝐵𝐹𝑣) = (𝑣𝐹𝐵)
2210, 18, 213eqtr3g 2856 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑣𝐹𝐵))
238, 22eqtr3d 2835 . . . . . . 7 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = (𝑣𝐹𝐵))
249, 19eqeltrdi 2898 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) ∈ 𝑆)
25 caovmo.dom . . . . . . . . . 10 dom 𝐹 = (𝑆 × 𝑆)
26 caovmo.3 . . . . . . . . . 10 ¬ ∅ ∈ 𝑆
2725, 26ndmovrcl 7315 . . . . . . . . 9 ((𝑢𝐹𝑤) ∈ 𝑆 → (𝑢𝑆𝑤𝑆))
2824, 27syl 17 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝑆𝑤𝑆))
29 oveq1 7142 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
30 id 22 . . . . . . . . . 10 (𝑥 = 𝑤𝑥 = 𝑤)
3129, 30eqeq12d 2814 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
32 caovmo.id . . . . . . . . 9 (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
3331, 32vtoclga 3522 . . . . . . . 8 (𝑤𝑆 → (𝑤𝐹𝐵) = 𝑤)
3428, 33simpl2im 507 . . . . . . 7 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = 𝑤)
357, 19eqeltrdi 2898 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) ∈ 𝑆)
3625, 26ndmovrcl 7315 . . . . . . . . 9 ((𝑢𝐹𝑣) ∈ 𝑆 → (𝑢𝑆𝑣𝑆))
3735, 36syl 17 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝑆𝑣𝑆))
38 oveq1 7142 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
39 id 22 . . . . . . . . . 10 (𝑥 = 𝑣𝑥 = 𝑣)
4038, 39eqeq12d 2814 . . . . . . . . 9 (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
4140, 32vtoclga 3522 . . . . . . . 8 (𝑣𝑆 → (𝑣𝐹𝐵) = 𝑣)
4237, 41simpl2im 507 . . . . . . 7 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑣𝐹𝐵) = 𝑣)
4323, 34, 423eqtr3d 2841 . . . . . 6 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
4443ax-gen 1797 . . . . 5 𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
456, 44mpgbir 1801 . . . 4 ∃*𝑤(𝑢𝐹𝑤) = 𝐵
463, 45vtoclg 3515 . . 3 (𝐴𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
47 moanimv 2681 . . 3 (∃*𝑤(𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝐴𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
4846, 47mpbir 234 . 2 ∃*𝑤(𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵)
49 eleq1 2877 . . . . . . 7 ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤) ∈ 𝑆𝐵𝑆))
5019, 49mpbiri 261 . . . . . 6 ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝐹𝑤) ∈ 𝑆)
5125, 26ndmovrcl 7315 . . . . . 6 ((𝐴𝐹𝑤) ∈ 𝑆 → (𝐴𝑆𝑤𝑆))
5250, 51syl 17 . . . . 5 ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝑆𝑤𝑆))
5352simpld 498 . . . 4 ((𝐴𝐹𝑤) = 𝐵𝐴𝑆)
5453ancri 553 . . 3 ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
5554moimi 2603 . 2 (∃*𝑤(𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
5648, 55ax-mp 5 1 ∃*𝑤(𝐴𝐹𝑤) = 𝐵
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃*wmo 2596  ∅c0 4243   × cxp 5517  dom cdm 5519  (class class class)co 7135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-dm 5529  df-iota 6283  df-fv 6332  df-ov 7138 This theorem is referenced by:  recmulnq  10377
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