Theorem sgrpass 13325

Description: A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)

Hypotheses

Ref	Expression
sgrpass.b	⊢ 𝐵 = (Base‘𝐺)
sgrpass.o	⊢ ⚬ = (+g‘𝐺)

Assertion

Ref	Expression
sgrpass	⊢ ((𝐺 ∈ Smgrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))

Proof of Theorem sgrpass

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

Step	Hyp	Ref	Expression
1		sgrpass.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		sgrpass.o	. . . 4 ⊢ ⚬ = (+g‘𝐺)
3	1, 2	issgrp 13320	. . 3 ⊢ (𝐺 ∈ Smgrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
4		oveq1 5969	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ 𝑦) = (𝑋 ⚬ 𝑦))
5	4	oveq1d 5977	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑦) ⚬ 𝑧))
6		oveq1 5969	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)))
7	5, 6	eqeq12d 2221	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
8		oveq2 5970	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ 𝑦) = (𝑋 ⚬ 𝑌))
9	8	oveq1d 5977	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑧))
10		oveq1 5969	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ⚬ 𝑧) = (𝑌 ⚬ 𝑧))
11	10	oveq2d 5978	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)))
12	9, 11	eqeq12d 2221	. . . . 5 ⊢ (𝑦 = 𝑌 → (((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
13		oveq2 5970	. . . . . 6 ⊢ (𝑧 = 𝑍 → ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑍))
14		oveq2 5970	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑌 ⚬ 𝑧) = (𝑌 ⚬ 𝑍))
15	14	oveq2d 5978	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑋 ⚬ (𝑌 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
16	13, 15	eqeq12d 2221	. . . . 5 ⊢ (𝑧 = 𝑍 → (((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
17	7, 12, 16	rspc3v 2897	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
18	17	com12 30	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
19	3, 18	simplbiim 387	. 2 ⊢ (𝐺 ∈ Smgrp → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
20	19	imp 124	1 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ‘cfv 5285  (class class class)co 5962  Basecbs 12917  +_gcplusg 12994  Mgmcmgm 13271  Smgrpcsgrp 13318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-ov 5965  df-inn 9067  df-2 9125  df-ndx 12920  df-slot 12921  df-base 12923  df-plusg 13007  df-sgrp 13319

This theorem is referenced by:  prdssgrpd  13332  mndass  13341  dfgrp2  13444  dfgrp3mlem  13515  dfgrp3me  13517  mulgnndir  13572  rngass  13786  rnglidlmsgrp  14344