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Theorem issgrpv 17202
Description: The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
Hypotheses
Ref Expression
issgrpn0.b 𝐵 = (Base‘𝑀)
issgrpn0.o = (+g𝑀)
Assertion
Ref Expression
issgrpv (𝑀𝑉 → (𝑀 ∈ SGrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem issgrpv
StepHypRef Expression
1 issgrpn0.b . . . 4 𝐵 = (Base‘𝑀)
2 issgrpn0.o . . . 4 = (+g𝑀)
31, 2ismgm 17159 . . 3 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
43anbi1d 740 . 2 (𝑀𝑉 → ((𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
51, 2issgrp 17201 . 2 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
6 r19.26-2 3063 . 2 (∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))) ↔ (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
74, 5, 63bitr4g 303 1 (𝑀𝑉 → (𝑀 ∈ SGrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  Mgmcmgm 17156  SGrpcsgrp 17199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-ov 6608  df-mgm 17158  df-sgrp 17200
This theorem is referenced by:  ismnd  17213  dfgrp2e  17364
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