Theorem sgrpplusgaopALT 49113

Description: Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sgrpplusgaopALT	⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺))

Proof of Theorem sgrpplusgaopALT

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝐺 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))) → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
2		eqid 2760	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		eqid 2760	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4	2, 3	issgrp 18825	. 2 ⊢ (𝐺 ∈ Smgrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
5		fvex 6892	. . 3 ⊢ (+g‘𝐺) ∈ V
6		fvex 6892	. . 3 ⊢ (Base‘𝐺) ∈ V
7		isasslaw 49110	. . 3 ⊢ (((+g‘𝐺) ∈ V ∧ (Base‘𝐺) ∈ V) → ((+g‘𝐺) assLaw (Base‘𝐺) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧))))
8	5, 6, 7	mp2an 705	. 2 ⊢ ((+g‘𝐺) assLaw (Base‘𝐺) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦)(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)(𝑦(+g‘𝐺)𝑧)))
9	1, 4, 8	3imtr4i 295	1 ⊢ (𝐺 ∈ Smgrp → (+g‘𝐺) assLaw (Base‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  +_gcplusg 17345  Mgmcmgm 18731  Smgrpcsgrp 18823   assLaw casslaw 49102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6489  df-fv 6541  df-ov 7417  df-sgrp 18824  df-asslaw 49106

This theorem is used by: (None)