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Theorem caovmo 6836
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 6622 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝐹𝑤) = (𝐴𝐹𝑤))
21eqeq1d 2623 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑤) = 𝐵))
32mobidv 2490 . . . 4 (𝑢 = 𝐴 → (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
4 oveq2 6623 . . . . . . 7 (𝑤 = 𝑣 → (𝑢𝐹𝑤) = (𝑢𝐹𝑣))
54eqeq1d 2623 . . . . . 6 (𝑤 = 𝑣 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝑢𝐹𝑣) = 𝐵))
65mo4 2516 . . . . 5 (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∀𝑤𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣))
7 simpr 477 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) = 𝐵)
87oveq2d 6631 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑤𝐹𝐵))
9 simpl 473 . . . . . . . . . 10 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) = 𝐵)
109oveq1d 6630 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → ((𝑢𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
11 vex 3193 . . . . . . . . . . 11 𝑢 ∈ V
12 vex 3193 . . . . . . . . . . 11 𝑤 ∈ V
13 vex 3193 . . . . . . . . . . 11 𝑣 ∈ V
14 caovmo.ass . . . . . . . . . . 11 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
1511, 12, 13, 14caovass 6799 . . . . . . . . . 10 ((𝑢𝐹𝑤)𝐹𝑣) = (𝑢𝐹(𝑤𝐹𝑣))
16 caovmo.com . . . . . . . . . . 11 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
1711, 12, 13, 16, 14caov12 6827 . . . . . . . . . 10 (𝑢𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝑢𝐹𝑣))
1815, 17eqtri 2643 . . . . . . . . 9 ((𝑢𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝑢𝐹𝑣))
19 caovmo.2 . . . . . . . . . . 11 𝐵𝑆
2019elexi 3203 . . . . . . . . . 10 𝐵 ∈ V
2120, 13, 16caovcom 6796 . . . . . . . . 9 (𝐵𝐹𝑣) = (𝑣𝐹𝐵)
2210, 18, 213eqtr3g 2678 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑣𝐹𝐵))
238, 22eqtr3d 2657 . . . . . . 7 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = (𝑣𝐹𝐵))
249, 19syl6eqel 2706 . . . . . . . . . 10 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) ∈ 𝑆)
25 caovmo.dom . . . . . . . . . . 11 dom 𝐹 = (𝑆 × 𝑆)
26 caovmo.3 . . . . . . . . . . 11 ¬ ∅ ∈ 𝑆
2725, 26ndmovrcl 6785 . . . . . . . . . 10 ((𝑢𝐹𝑤) ∈ 𝑆 → (𝑢𝑆𝑤𝑆))
2824, 27syl 17 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝑆𝑤𝑆))
2928simprd 479 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤𝑆)
30 oveq1 6622 . . . . . . . . . 10 (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
31 id 22 . . . . . . . . . 10 (𝑥 = 𝑤𝑥 = 𝑤)
3230, 31eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
33 caovmo.id . . . . . . . . 9 (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
3432, 33vtoclga 3262 . . . . . . . 8 (𝑤𝑆 → (𝑤𝐹𝐵) = 𝑤)
3529, 34syl 17 . . . . . . 7 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = 𝑤)
367, 19syl6eqel 2706 . . . . . . . . . 10 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) ∈ 𝑆)
3725, 26ndmovrcl 6785 . . . . . . . . . 10 ((𝑢𝐹𝑣) ∈ 𝑆 → (𝑢𝑆𝑣𝑆))
3836, 37syl 17 . . . . . . . . 9 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝑆𝑣𝑆))
3938simprd 479 . . . . . . . 8 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑣𝑆)
40 oveq1 6622 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
41 id 22 . . . . . . . . . 10 (𝑥 = 𝑣𝑥 = 𝑣)
4240, 41eqeq12d 2636 . . . . . . . . 9 (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
4342, 33vtoclga 3262 . . . . . . . 8 (𝑣𝑆 → (𝑣𝐹𝐵) = 𝑣)
4439, 43syl 17 . . . . . . 7 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑣𝐹𝐵) = 𝑣)
4523, 35, 443eqtr3d 2663 . . . . . 6 (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
4645ax-gen 1719 . . . . 5 𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
476, 46mpgbir 1723 . . . 4 ∃*𝑤(𝑢𝐹𝑤) = 𝐵
483, 47vtoclg 3256 . . 3 (𝐴𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
49 moanimv 2530 . . 3 (∃*𝑤(𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝐴𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
5048, 49mpbir 221 . 2 ∃*𝑤(𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵)
51 eleq1 2686 . . . . . . 7 ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤) ∈ 𝑆𝐵𝑆))
5219, 51mpbiri 248 . . . . . 6 ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝐹𝑤) ∈ 𝑆)
5325, 26ndmovrcl 6785 . . . . . 6 ((𝐴𝐹𝑤) ∈ 𝑆 → (𝐴𝑆𝑤𝑆))
5452, 53syl 17 . . . . 5 ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝑆𝑤𝑆))
5554simpld 475 . . . 4 ((𝐴𝐹𝑤) = 𝐵𝐴𝑆)
5655ancri 574 . . 3 ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
5756moimi 2519 . 2 (∃*𝑤(𝐴𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
5850, 57ax-mp 5 1 ∃*𝑤(𝐴𝐹𝑤) = 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  ∃*wmo 2470  c0 3897   × cxp 5082  dom cdm 5084  (class class class)co 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-dm 5094  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  recmulnq  9746
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