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Theorem sgrpass 17676
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 17671 . . 3 (𝐺 ∈ SGrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4 oveq1 6929 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
54oveq1d 6937 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
6 oveq1 6929 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
75, 6eqeq12d 2792 . . . . 5 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
8 oveq2 6930 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
98oveq1d 6937 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
10 oveq1 6929 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1110oveq2d 6938 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
129, 11eqeq12d 2792 . . . . 5 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
13 oveq2 6930 . . . . . 6 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
14 oveq2 6930 . . . . . . 7 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
1514oveq2d 6938 . . . . . 6 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
1613, 15eqeq12d 2792 . . . . 5 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
177, 12, 16rspc3v 3526 . . . 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 500 . 2 (𝐺 ∈ SGrp → ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2019imp 397 1 ((𝐺 ∈ SGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2106  wral 3089  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  Mgmcmgm 17626  SGrpcsgrp 17669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-sgrp 17670
This theorem is referenced by:  mndass  17688  dfgrp2  17834  dfgrp3lem  17900  dfgrp3e  17902  mulgnndir  17955
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