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Theorem issgrp 17108
 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 fvex 6113 . . . 4 (Base‘𝑔) ∈ V
21a1i 11 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
3 fveq2 6103 . . . 4 (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
4 issgrp.b . . . 4 𝐵 = (Base‘𝑀)
53, 4syl6eqr 2662 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
6 fvex 6113 . . . . 5 (+g𝑔) ∈ V
76a1i 11 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) ∈ V)
8 fveq2 6103 . . . . . 6 (𝑔 = 𝑀 → (+g𝑔) = (+g𝑀))
98adantr 480 . . . . 5 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = (+g𝑀))
10 issgrp.o . . . . 5 = (+g𝑀)
119, 10syl6eqr 2662 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = )
12 simplr 788 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
13 id 22 . . . . . . . . . 10 (𝑜 = 𝑜 = )
14 oveq 6555 . . . . . . . . . 10 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
15 eqidd 2611 . . . . . . . . . 10 (𝑜 = 𝑧 = 𝑧)
1613, 14, 15oveq123d 6570 . . . . . . . . 9 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
17 eqidd 2611 . . . . . . . . . 10 (𝑜 = 𝑥 = 𝑥)
18 oveq 6555 . . . . . . . . . 10 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
1913, 17, 18oveq123d 6570 . . . . . . . . 9 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
2016, 19eqeq12d 2625 . . . . . . . 8 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2120adantl 481 . . . . . . 7 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2212, 21raleqbidv 3129 . . . . . 6 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2312, 22raleqbidv 3129 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2412, 23raleqbidv 3129 . . . 4 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
257, 11, 24sbcied2 3440 . . 3 ((𝑔 = 𝑀𝑏 = 𝐵) → ([(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
262, 5, 25sbcied2 3440 . 2 (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
27 df-sgrp 17107 . 2 SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
2826, 27elrab2 3333 1 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063  SGrpcsgrp 17106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-sgrp 17107 This theorem is referenced by:  issgrpv  17109  issgrpn0  17110  isnsgrp  17111  sgrpmgm  17112  sgrpass  17113  sgrp0  17114  sgrp0b  17115  sgrp1  17116  sgrp2nmndlem4  17238  copissgrp  41598  nnsgrp  41607  sgrpplusgaopALT  41621  sgrp2sgrp  41654  lidlmsgrp  41716  2zrngasgrp  41730  2zrngmsgrp  41737
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