Theorem sgrpplusgaopALT 43406

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 ∈ SGrp → (+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 477	. 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 2772	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
3		eqid 2772	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
4	2, 3	issgrp 17743	. 2 ⊢ (𝐺 ∈ SGrp ↔ (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 6506	. . 3 ⊢ (+g‘𝐺) ∈ V
6		fvex 6506	. . 3 ⊢ (Base‘𝐺) ∈ V
7		isasslaw 43403	. . 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 679	. 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 284	1 ⊢ (𝐺 ∈ SGrp → (+g‘𝐺) assLaw (Base‘𝐺))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082  Vcvv 3409   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970  Basecbs 16329  +_gcplusg 16411  Mgmcmgm 17698  SGrpcsgrp 17741   assLaw casslaw 43395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-iota 6146  df-fv 6190  df-ov 6973  df-sgrp 17742  df-asslaw 43399

This theorem is referenced by: (None)