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Theorem caovimo 5957
Description: Uniqueness of inverse element in commutative, associative operation with identity. The identity element is 𝐵. (Contributed by Jim Kingdon, 18-Sep-2019.)
Hypotheses
Ref Expression
caovimo.idel 𝐵𝑆
caovimo.com ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovimo.ass ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovimo.id (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
Assertion
Ref Expression
caovimo (𝐴𝑆 → ∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
Distinct variable groups:   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧
Allowed substitution hint:   𝐵(𝑧)

Proof of Theorem caovimo
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 oveq1 5774 . . . . . . 7 ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
21adantl 275 . . . . . 6 ((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
323ad2ant2 1003 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
4 df-3an 964 . . . . . . . . 9 ((𝐴𝑆𝑤𝑆𝑣𝑆) ↔ ((𝐴𝑆𝑤𝑆) ∧ 𝑣𝑆))
5 caovimo.ass . . . . . . . . . . . . . 14 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
65adantl 275 . . . . . . . . . . . . 13 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7 simp1 981 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝐴𝑆)
8 simp2 982 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝑤𝑆)
9 simp3 983 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝑣𝑆)
106, 7, 8, 9caovassd 5923 . . . . . . . . . . . 12 ((𝐴𝑆𝑤𝑆𝑣𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐴𝐹(𝑤𝐹𝑣)))
11 caovimo.com . . . . . . . . . . . . . 14 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
1211adantl 275 . . . . . . . . . . . . 13 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
137, 8, 9, 12, 6caov12d 5945 . . . . . . . . . . . 12 ((𝐴𝑆𝑤𝑆𝑣𝑆) → (𝐴𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝐴𝐹𝑣)))
1410, 13eqtrd 2170 . . . . . . . . . . 11 ((𝐴𝑆𝑤𝑆𝑣𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
1514adantr 274 . . . . . . . . . 10 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
16 oveq2 5775 . . . . . . . . . . . 12 ((𝐴𝐹𝑣) = 𝐵 → (𝑤𝐹(𝐴𝐹𝑣)) = (𝑤𝐹𝐵))
17 oveq1 5774 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
18 id 19 . . . . . . . . . . . . . 14 (𝑥 = 𝑤𝑥 = 𝑤)
1917, 18eqeq12d 2152 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
20 caovimo.id . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
2119, 20vtoclga 2747 . . . . . . . . . . . 12 (𝑤𝑆 → (𝑤𝐹𝐵) = 𝑤)
2216, 21sylan9eqr 2192 . . . . . . . . . . 11 ((𝑤𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
23223ad2antl2 1144 . . . . . . . . . 10 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
2415, 23eqtrd 2170 . . . . . . . . 9 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
254, 24sylanbr 283 . . . . . . . 8 ((((𝐴𝑆𝑤𝑆) ∧ 𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
2625anasss 396 . . . . . . 7 (((𝐴𝑆𝑤𝑆) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
27263impa 1176 . . . . . 6 ((𝐴𝑆𝑤𝑆 ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
28273adant2r 1211 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
2911adantl 275 . . . . . . . . 9 ((𝑣𝑆 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
30 caovimo.idel . . . . . . . . . 10 𝐵𝑆
3130a1i 9 . . . . . . . . 9 (𝑣𝑆𝐵𝑆)
32 id 19 . . . . . . . . 9 (𝑣𝑆𝑣𝑆)
3329, 31, 32caovcomd 5920 . . . . . . . 8 (𝑣𝑆 → (𝐵𝐹𝑣) = (𝑣𝐹𝐵))
34 oveq1 5774 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
35 id 19 . . . . . . . . . 10 (𝑥 = 𝑣𝑥 = 𝑣)
3634, 35eqeq12d 2152 . . . . . . . . 9 (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
3736, 20vtoclga 2747 . . . . . . . 8 (𝑣𝑆 → (𝑣𝐹𝐵) = 𝑣)
3833, 37eqtrd 2170 . . . . . . 7 (𝑣𝑆 → (𝐵𝐹𝑣) = 𝑣)
3938adantr 274 . . . . . 6 ((𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝐵𝐹𝑣) = 𝑣)
40393ad2ant3 1004 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → (𝐵𝐹𝑣) = 𝑣)
413, 28, 403eqtr3d 2178 . . . 4 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣)
42413expib 1184 . . 3 (𝐴𝑆 → (((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
4342alrimivv 1847 . 2 (𝐴𝑆 → ∀𝑤𝑣(((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
44 eleq1 2200 . . . 4 (𝑤 = 𝑣 → (𝑤𝑆𝑣𝑆))
45 oveq2 5775 . . . . 5 (𝑤 = 𝑣 → (𝐴𝐹𝑤) = (𝐴𝐹𝑣))
4645eqeq1d 2146 . . . 4 (𝑤 = 𝑣 → ((𝐴𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑣) = 𝐵))
4744, 46anbi12d 464 . . 3 (𝑤 = 𝑣 → ((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)))
4847mo4 2058 . 2 (∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ ∀𝑤𝑣(((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
4943, 48sylibr 133 1 (𝐴𝑆 → ∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962  wal 1329   = wceq 1331  wcel 1480  ∃*wmo 1998  (class class class)co 5767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  recmulnqg  7192
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