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Theorem sgrpass 18688
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 ∈ Smgrp ∧ (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 18683 . . 3 (𝐺 ∈ Smgrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4 oveq1 7426 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
54oveq1d 7434 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
6 oveq1 7426 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
75, 6eqeq12d 2741 . . . . 5 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
8 oveq2 7427 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
98oveq1d 7434 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
10 oveq1 7426 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1110oveq2d 7435 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
129, 11eqeq12d 2741 . . . . 5 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
13 oveq2 7427 . . . . . 6 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
14 oveq2 7427 . . . . . . 7 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
1514oveq2d 7435 . . . . . 6 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
1613, 15eqeq12d 2741 . . . . 5 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
177, 12, 16rspc3v 3622 . . . 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 503 . 2 (𝐺 ∈ Smgrp → ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2019imp 405 1 ((𝐺 ∈ Smgrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  Mgmcmgm 18601  Smgrpcsgrp 18681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-sgrp 18682
This theorem is referenced by:  prdssgrpd  18696  mndass  18706  gsumsgrpccat  18800  dfgrp2  18927  dfgrp3lem  19002  dfgrp3e  19004  mulgnndir  19066  cntzsgrpcl  19297  rngass  20111  rnglidlmsgrp  21153
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