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Theorem issgrp 18733
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
issgrp.b 𝐵 = (Base‘𝑀)
issgrp.o = (+g𝑀)
Assertion
Ref Expression
issgrp (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem issgrp
Dummy variables 𝑏 𝑔 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6921 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
2 fveq2 6906 . . . 4 (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
3 issgrp.b . . . 4 𝐵 = (Base‘𝑀)
42, 3eqtr4di 2795 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
5 fvexd 6921 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) ∈ V)
6 fveq2 6906 . . . . . 6 (𝑔 = 𝑀 → (+g𝑔) = (+g𝑀))
76adantr 480 . . . . 5 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = (+g𝑀))
8 issgrp.o . . . . 5 = (+g𝑀)
97, 8eqtr4di 2795 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = )
10 simplr 769 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 id 22 . . . . . . . . . 10 (𝑜 = 𝑜 = )
12 oveq 7437 . . . . . . . . . 10 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
13 eqidd 2738 . . . . . . . . . 10 (𝑜 = 𝑧 = 𝑧)
1411, 12, 13oveq123d 7452 . . . . . . . . 9 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
15 eqidd 2738 . . . . . . . . . 10 (𝑜 = 𝑥 = 𝑥)
16 oveq 7437 . . . . . . . . . 10 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
1711, 15, 16oveq123d 7452 . . . . . . . . 9 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
1814, 17eqeq12d 2753 . . . . . . . 8 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1918adantl 481 . . . . . . 7 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2010, 19raleqbidv 3346 . . . . . 6 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2110, 20raleqbidv 3346 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2210, 21raleqbidv 3346 . . . 4 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
235, 9, 22sbcied2 3833 . . 3 ((𝑔 = 𝑀𝑏 = 𝐵) → ([(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
241, 4, 23sbcied2 3833 . 2 (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
25 df-sgrp 18732 . 2 Smgrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
2624, 25elrab2 3695 1 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  [wsbc 3788  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mgmcmgm 18651  Smgrpcsgrp 18731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-sgrp 18732
This theorem is referenced by:  issgrpv  18734  issgrpn0  18735  isnsgrp  18736  sgrpmgm  18737  sgrpass  18738  sgrp0  18740  sgrp0b  18741  sgrp1  18742  efmndsgrp  18899  smndex1sgrp  18921  sgrp2nmndlem4  18941  rnglidlmsgrp  21256  copissgrp  48084  nnsgrp  48093  sgrpplusgaopALT  48111  sgrp2sgrp  48144  2zrngasgrp  48162  2zrngmsgrp  48169
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