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Theorem caovimo 5776
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 5601 . . . . . . 7 ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
21adantl 271 . . . . . 6 ((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
323ad2ant2 963 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
4 df-3an 924 . . . . . . . . 9 ((𝐴𝑆𝑤𝑆𝑣𝑆) ↔ ((𝐴𝑆𝑤𝑆) ∧ 𝑣𝑆))
5 caovimo.ass . . . . . . . . . . . . . 14 ((𝑥𝑆𝑦𝑆𝑧𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
65adantl 271 . . . . . . . . . . . . 13 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7 simp1 941 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝐴𝑆)
8 simp2 942 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝑤𝑆)
9 simp3 943 . . . . . . . . . . . . 13 ((𝐴𝑆𝑤𝑆𝑣𝑆) → 𝑣𝑆)
106, 7, 8, 9caovassd 5742 . . . . . . . . . . . 12 ((𝐴𝑆𝑤𝑆𝑣𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐴𝐹(𝑤𝐹𝑣)))
11 caovimo.com . . . . . . . . . . . . . 14 ((𝑥𝑆𝑦𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
1211adantl 271 . . . . . . . . . . . . 13 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
137, 8, 9, 12, 6caov12d 5764 . . . . . . . . . . . 12 ((𝐴𝑆𝑤𝑆𝑣𝑆) → (𝐴𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝐴𝐹𝑣)))
1410, 13eqtrd 2117 . . . . . . . . . . 11 ((𝐴𝑆𝑤𝑆𝑣𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
1514adantr 270 . . . . . . . . . 10 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
16 oveq2 5602 . . . . . . . . . . . 12 ((𝐴𝐹𝑣) = 𝐵 → (𝑤𝐹(𝐴𝐹𝑣)) = (𝑤𝐹𝐵))
17 oveq1 5601 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
18 id 19 . . . . . . . . . . . . . 14 (𝑥 = 𝑤𝑥 = 𝑤)
1917, 18eqeq12d 2099 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
20 caovimo.id . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑥𝐹𝐵) = 𝑥)
2119, 20vtoclga 2677 . . . . . . . . . . . 12 (𝑤𝑆 → (𝑤𝐹𝐵) = 𝑤)
2216, 21sylan9eqr 2139 . . . . . . . . . . 11 ((𝑤𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
23223ad2antl2 1104 . . . . . . . . . 10 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
2415, 23eqtrd 2117 . . . . . . . . 9 (((𝐴𝑆𝑤𝑆𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
254, 24sylanbr 279 . . . . . . . 8 ((((𝐴𝑆𝑤𝑆) ∧ 𝑣𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
2625anasss 391 . . . . . . 7 (((𝐴𝑆𝑤𝑆) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
27263impa 1136 . . . . . 6 ((𝐴𝑆𝑤𝑆 ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
28273adant2r 1167 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
2911adantl 271 . . . . . . . . 9 ((𝑣𝑆 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
30 caovimo.idel . . . . . . . . . 10 𝐵𝑆
3130a1i 9 . . . . . . . . 9 (𝑣𝑆𝐵𝑆)
32 id 19 . . . . . . . . 9 (𝑣𝑆𝑣𝑆)
3329, 31, 32caovcomd 5739 . . . . . . . 8 (𝑣𝑆 → (𝐵𝐹𝑣) = (𝑣𝐹𝐵))
34 oveq1 5601 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
35 id 19 . . . . . . . . . 10 (𝑥 = 𝑣𝑥 = 𝑣)
3634, 35eqeq12d 2099 . . . . . . . . 9 (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
3736, 20vtoclga 2677 . . . . . . . 8 (𝑣𝑆 → (𝑣𝐹𝐵) = 𝑣)
3833, 37eqtrd 2117 . . . . . . 7 (𝑣𝑆 → (𝐵𝐹𝑣) = 𝑣)
3938adantr 270 . . . . . 6 ((𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝐵𝐹𝑣) = 𝑣)
40393ad2ant3 964 . . . . 5 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → (𝐵𝐹𝑣) = 𝑣)
413, 28, 403eqtr3d 2125 . . . 4 ((𝐴𝑆 ∧ (𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣)
42413expib 1144 . . 3 (𝐴𝑆 → (((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
4342alrimivv 1800 . 2 (𝐴𝑆 → ∀𝑤𝑣(((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
44 eleq1 2147 . . . 4 (𝑤 = 𝑣 → (𝑤𝑆𝑣𝑆))
45 oveq2 5602 . . . . 5 (𝑤 = 𝑣 → (𝐴𝐹𝑤) = (𝐴𝐹𝑣))
4645eqeq1d 2093 . . . 4 (𝑤 = 𝑣 → ((𝐴𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑣) = 𝐵))
4744, 46anbi12d 457 . . 3 (𝑤 = 𝑣 → ((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)))
4847mo4 2006 . 2 (∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ ∀𝑤𝑣(((𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
4943, 48sylibr 132 1 (𝐴𝑆 → ∃*𝑤(𝑤𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 922  wal 1285   = wceq 1287  wcel 1436  ∃*wmo 1946  (class class class)co 5594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-iota 4937  df-fv 4980  df-ov 5597
This theorem is referenced by:  recmulnqg  6871
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