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Theorem sgrpass 12706
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 12701 . . 3 (𝐺 ∈ Smgrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
4 oveq1 5876 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 𝑦) = (𝑋 𝑦))
54oveq1d 5884 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 𝑦) 𝑧) = ((𝑋 𝑦) 𝑧))
6 oveq1 5876 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 𝑧)) = (𝑋 (𝑦 𝑧)))
75, 6eqeq12d 2192 . . . . 5 (𝑥 = 𝑋 → (((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) ↔ ((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧))))
8 oveq2 5877 . . . . . . 7 (𝑦 = 𝑌 → (𝑋 𝑦) = (𝑋 𝑌))
98oveq1d 5884 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 𝑦) 𝑧) = ((𝑋 𝑌) 𝑧))
10 oveq1 5876 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 𝑧) = (𝑌 𝑧))
1110oveq2d 5885 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 𝑧)) = (𝑋 (𝑌 𝑧)))
129, 11eqeq12d 2192 . . . . 5 (𝑦 = 𝑌 → (((𝑋 𝑦) 𝑧) = (𝑋 (𝑦 𝑧)) ↔ ((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧))))
13 oveq2 5877 . . . . . 6 (𝑧 = 𝑍 → ((𝑋 𝑌) 𝑧) = ((𝑋 𝑌) 𝑍))
14 oveq2 5877 . . . . . . 7 (𝑧 = 𝑍 → (𝑌 𝑧) = (𝑌 𝑍))
1514oveq2d 5885 . . . . . 6 (𝑧 = 𝑍 → (𝑋 (𝑌 𝑧)) = (𝑋 (𝑌 𝑍)))
1613, 15eqeq12d 2192 . . . . 5 (𝑧 = 𝑍 → (((𝑋 𝑌) 𝑧) = (𝑋 (𝑌 𝑧)) ↔ ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
177, 12, 16rspc3v 2857 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
1817com12 30 . . 3 (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)) → ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
193, 18simplbiim 387 . 2 (𝐺 ∈ Smgrp → ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))
2019imp 124 1 ((𝐺 ∈ Smgrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  cfv 5212  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  Mgmcmgm 12665  Smgrpcsgrp 12699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-sgrp 12700
This theorem is referenced by:  mndass  12717  dfgrp2  12792  dfgrp3mlem  12857  dfgrp3me  12859  mulgnndir  12900
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