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Theorem sgrpass 17483
Description: A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
Hypotheses
Ref Expression
sgrpass.b 𝐵 = (Base‘𝐺)
sgrpass.o = (+g𝐺)
Assertion
Ref Expression
sgrpass ((𝐺 ∈ SGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Proof of Theorem sgrpass
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sgrpass.b . . . 4 𝐵 = (Base‘𝐺)
2 sgrpass.o . . . 4 = (+g𝐺)
31, 2issgrp 17478 . . 3 (𝐺 ∈ SGrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4 oveq1 6812 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
54oveq1d 6820 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
6 oveq1 6812 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
75, 6eqeq12d 2767 . . . . 5 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
8 oveq2 6813 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
98oveq1d 6820 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
10 oveq1 6812 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1110oveq2d 6821 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
129, 11eqeq12d 2767 . . . . 5 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
13 oveq2 6813 . . . . . 6 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
14 oveq2 6813 . . . . . . 7 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
1514oveq2d 6821 . . . . . 6 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
1613, 15eqeq12d 2767 . . . . 5 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
177, 12, 16rspc3v 3456 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
1817com12 32 . . 3 (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
193, 18simplbiim 661 . 2 (𝐺 ∈ SGrp → ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2019imp 444 1 ((𝐺 ∈ SGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  cfv 6041  (class class class)co 6805  Basecbs 16051  +gcplusg 16135  Mgmcmgm 17433  SGrpcsgrp 17476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-nul 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-iota 6004  df-fv 6049  df-ov 6808  df-sgrp 17477
This theorem is referenced by:  mndass  17495  dfgrp2  17640  dfgrp3lem  17706  dfgrp3e  17708  mulgnndir  17762
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