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Theorem caovassg 5659
 Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013.) (Revised by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovassg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
Assertion
Ref Expression
caovassg ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovassg
StepHypRef Expression
1 caovassg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
21ralrimivvva 2402 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
3 oveq1 5519 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
43oveq1d 5527 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐹𝑧) = ((𝐴𝐹𝑦)𝐹𝑧))
5 oveq1 5519 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹(𝑦𝐹𝑧)) = (𝐴𝐹(𝑦𝐹𝑧)))
64, 5eqeq12d 2054 . . 3 (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) ↔ ((𝐴𝐹𝑦)𝐹𝑧) = (𝐴𝐹(𝑦𝐹𝑧))))
7 oveq2 5520 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
87oveq1d 5527 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐹𝑧) = ((𝐴𝐹𝐵)𝐹𝑧))
9 oveq1 5519 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
109oveq2d 5528 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹(𝑦𝐹𝑧)) = (𝐴𝐹(𝐵𝐹𝑧)))
118, 10eqeq12d 2054 . . 3 (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐹𝑧) = (𝐴𝐹(𝑦𝐹𝑧)) ↔ ((𝐴𝐹𝐵)𝐹𝑧) = (𝐴𝐹(𝐵𝐹𝑧))))
12 oveq2 5520 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐹𝑧) = ((𝐴𝐹𝐵)𝐹𝐶))
13 oveq2 5520 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
1413oveq2d 5528 . . . 4 (𝑧 = 𝐶 → (𝐴𝐹(𝐵𝐹𝑧)) = (𝐴𝐹(𝐵𝐹𝐶)))
1512, 14eqeq12d 2054 . . 3 (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐹𝑧) = (𝐴𝐹(𝐵𝐹𝑧)) ↔ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))))
166, 11, 15rspc3v 2665 . 2 ((𝐴𝑆𝐵𝑆𝐶𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))))
172, 16mpan9 265 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306  (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-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:  caovassd  5660  caovass  5661  grprinvlem  5695  grprinvd  5696  grpridd  5697  iseqsplit  9238  iseqcaopr  9242
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