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Theorem caovimo 5694
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 5519 . . . . . . 7 ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
21adantl 262 . . . . . 6 ((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
323ad2ant2 926 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
4 df-3an 887 . . . . . . . . 9 ((𝐴𝑆𝑤𝑆𝑣𝑆) ↔ ((𝐴𝑆𝑤𝑆) ∧ 𝑣𝑆))
5 caovimo.ass . . . . . . . . . . . . . 14 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
65adantl 262 . . . . . . . . . . . . 13 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7 simp1 904 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝐴𝑆)
8 simp2 905 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝑤𝑆)
9 simp3 906 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝑣𝑆)
106, 7, 8, 9caovassd 5660 . . . . . . . . . . . 12 ((𝐴𝑆𝑤𝑆𝑣𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐴𝐹(𝑤𝐹𝑣)))
11 caovimo.com . . . . . . . . . . . . . 14 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
1211adantl 262 . . . . . . . . . . . . 13 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
137, 8, 9, 12, 6caov12d 5682 . . . . . . . . . . . 12 ((𝐴𝑆𝑤𝑆𝑣𝑆) → (𝐴𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝐴𝐹𝑣)))
1410, 13eqtrd 2072 . . . . . . . . . . 11 ((𝐴𝑆𝑤𝑆𝑣𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
1514adantr 261 . . . . . . . . . 10 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
16 oveq2 5520 . . . . . . . . . . . 12 ((𝐴𝐹𝑣) = 𝐵 → (𝑤𝐹(𝐴𝐹𝑣)) = (𝑤𝐹𝐵))
17 oveq1 5519 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
18 id 19 . . . . . . . . . . . . . 14 (𝑥 = 𝑤𝑥 = 𝑤)
1917, 18eqeq12d 2054 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
20 caovimo.id . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
2119, 20vtoclga 2619 . . . . . . . . . . . 12 (𝑤𝑆 → (𝑤𝐹𝐵) = 𝑤)
2216, 21sylan9eqr 2094 . . . . . . . . . . 11 ((𝑤𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
23223ad2antl2 1067 . . . . . . . . . 10 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
2415, 23eqtrd 2072 . . . . . . . . 9 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
254, 24sylanbr 269 . . . . . . . 8 ((((𝐴𝑆𝑤𝑆) ∧ 𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
2625anasss 379 . . . . . . 7 (((𝐴𝑆𝑤𝑆) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
27263impa 1099 . . . . . 6 ((𝐴𝑆𝑤𝑆 ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
28273adant2r 1130 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
2911adantl 262 . . . . . . . . 9 ((𝑣𝑆 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
30 caovimo.idel . . . . . . . . . 10 𝐵𝑆
3130a1i 9 . . . . . . . . 9 (𝑣𝑆𝐵𝑆)
32 id 19 . . . . . . . . 9 (𝑣𝑆𝑣𝑆)
3329, 31, 32caovcomd 5657 . . . . . . . 8 (𝑣𝑆 → (𝐵𝐹𝑣) = (𝑣𝐹𝐵))
34 oveq1 5519 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
35 id 19 . . . . . . . . . 10 (𝑥 = 𝑣𝑥 = 𝑣)
3634, 35eqeq12d 2054 . . . . . . . . 9 (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
3736, 20vtoclga 2619 . . . . . . . 8 (𝑣𝑆 → (𝑣𝐹𝐵) = 𝑣)
3833, 37eqtrd 2072 . . . . . . 7 (𝑣𝑆 → (𝐵𝐹𝑣) = 𝑣)
3938adantr 261 . . . . . 6 ((𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝐵𝐹𝑣) = 𝑣)
40393ad2ant3 927 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → (𝐵𝐹𝑣) = 𝑣)
413, 28, 403eqtr3d 2080 . . . 4 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣)
42413expib 1107 . . 3 (𝐴𝑆 → (((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
4342alrimivv 1755 . 2 (𝐴𝑆 → ∀𝑤𝑣(((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
44 eleq1 2100 . . . 4 (𝑤 = 𝑣 → (𝑤𝑆𝑣𝑆))
45 oveq2 5520 . . . . 5 (𝑤 = 𝑣 → (𝐴𝐹𝑤) = (𝐴𝐹𝑣))
4645eqeq1d 2048 . . . 4 (𝑤 = 𝑣 → ((𝐴𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑣) = 𝐵))
4744, 46anbi12d 442 . . 3 (𝑤 = 𝑣 → ((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)))
4847mo4 1961 . 2 (∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ ∀𝑤𝑣(((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
4943, 48sylibr 137 1 (𝐴𝑆 → ∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885  wal 1241   = wceq 1243  wcel 1393  ∃*wmo 1901  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by:  recmulnqg  6487
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