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Theorem issgrp 17724
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
issgrp.b 𝐵 = (Base‘𝑀)
issgrp.o = (+g𝑀)
Assertion
Ref Expression
issgrp (𝑀 ∈ SGrp ↔ (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 6556 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
2 fveq2 6541 . . . 4 (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
3 issgrp.b . . . 4 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2848 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
5 fvexd 6556 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) ∈ V)
6 fveq2 6541 . . . . . 6 (𝑔 = 𝑀 → (+g𝑔) = (+g𝑀))
76adantr 481 . . . . 5 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = (+g𝑀))
8 issgrp.o . . . . 5 = (+g𝑀)
97, 8syl6eqr 2848 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = )
10 simplr 765 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 id 22 . . . . . . . . . 10 (𝑜 = 𝑜 = )
12 oveq 7025 . . . . . . . . . 10 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
13 eqidd 2795 . . . . . . . . . 10 (𝑜 = 𝑧 = 𝑧)
1411, 12, 13oveq123d 7040 . . . . . . . . 9 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
15 eqidd 2795 . . . . . . . . . 10 (𝑜 = 𝑥 = 𝑥)
16 oveq 7025 . . . . . . . . . 10 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
1711, 15, 16oveq123d 7040 . . . . . . . . 9 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
1814, 17eqeq12d 2809 . . . . . . . 8 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1918adantl 482 . . . . . . 7 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2010, 19raleqbidv 3360 . . . . . 6 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2110, 20raleqbidv 3360 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2210, 21raleqbidv 3360 . . . 4 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
235, 9, 22sbcied2 3745 . . 3 ((𝑔 = 𝑀𝑏 = 𝐵) → ([(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
241, 4, 23sbcied2 3745 . 2 (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
25 df-sgrp 17723 . 2 SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
2624, 25elrab2 3620 1 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1522  wcel 2080  wral 3104  Vcvv 3436  [wsbc 3707  cfv 6228  (class class class)co 7019  Basecbs 16312  +gcplusg 16394  Mgmcmgm 17679  SGrpcsgrp 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768  ax-nul 5104
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-iota 6192  df-fv 6236  df-ov 7022  df-sgrp 17723
This theorem is referenced by:  issgrpv  17725  issgrpn0  17726  isnsgrp  17727  sgrpmgm  17728  sgrpass  17729  sgrp0  17730  sgrp0b  17731  sgrp1  17732  sgrp2nmndlem4  17854  copissgrp  43571  nnsgrp  43580  sgrpplusgaopALT  43594  sgrp2sgrp  43627  lidlmsgrp  43689  2zrngasgrp  43703  2zrngmsgrp  43710
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