Theorem sgrpass 13706

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 13701	. . 3 ⊢ (𝐺 ∈ Smgrp ↔ (𝐺 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
4		oveq1 6086	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ 𝑦) = (𝑋 ⚬ 𝑦))
5	4	oveq1d 6094	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑦) ⚬ 𝑧))
6		oveq1 6086	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)))
7	5, 6	eqeq12d 2253	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
8		oveq2 6087	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ 𝑦) = (𝑋 ⚬ 𝑌))
9	8	oveq1d 6094	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑧))
10		oveq1 6086	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑦 ⚬ 𝑧) = (𝑌 ⚬ 𝑧))
11	10	oveq2d 6095	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)))
12	9, 11	eqeq12d 2253	. . . . 5 ⊢ (𝑦 = 𝑌 → (((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
13		oveq2 6087	. . . . . 6 ⊢ (𝑧 = 𝑍 → ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑍))
14		oveq2 6087	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑌 ⚬ 𝑧) = (𝑌 ⚬ 𝑍))
15	14	oveq2d 6095	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑋 ⚬ (𝑌 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
16	13, 15	eqeq12d 2253	. . . . 5 ⊢ (𝑧 = 𝑍 → (((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
17	7, 12, 16	rspc3v 2946	. . . 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 391	. 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 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  Mgmcmgm 13657  Smgrpcsgrp 13699

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-inn 9288  df-2 9346  df-ndx 13338  df-slot 13339  df-base 13341  df-plusg 13427  df-sgrp 13700

This theorem is referenced by:  mndass  13720  dfgrp2  13815  dfgrp3mlem  13886  dfgrp3me  13888  mulgnndir  13937  prdssgrpd  14174  rngass  14221  rnglidlmsgrp  14817