Theorem isass 38047

Description: The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isass.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
isass	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))

Distinct variable groups:   𝑥,𝐺,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem isass

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5852	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2	1	dmeqd 5854	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3	2	eleq2d 2822	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ dom dom 𝑔 ↔ 𝑥 ∈ dom dom 𝐺))
4	2	eleq2d 2822	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦 ∈ dom dom 𝑔 ↔ 𝑦 ∈ dom dom 𝐺))
5	2	eleq2d 2822	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑧 ∈ dom dom 𝑔 ↔ 𝑧 ∈ dom dom 𝐺))
6	3, 4, 5	3anbi123d 1438	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑥 ∈ dom dom 𝑔 ∧ 𝑦 ∈ dom dom 𝑔 ∧ 𝑧 ∈ dom dom 𝑔) ↔ (𝑥 ∈ dom dom 𝐺 ∧ 𝑦 ∈ dom dom 𝐺 ∧ 𝑧 ∈ dom dom 𝐺)))
7		oveq 7364	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8	7	oveq1d 7373	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝑔𝑧))
9		oveq 7364	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝐺𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
10	8, 9	eqtrd 2771	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
11		oveq 7364	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
12	11	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝑔(𝑦𝐺𝑧)))
13		oveq 7364	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝐺𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
14	12, 13	eqtrd 2771	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
15	10, 14	eqeq12d 2752	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
16	6, 15	imbi12d 344	. . . . . 6 ⊢ (𝑔 = 𝐺 → (((𝑥 ∈ dom dom 𝑔 ∧ 𝑦 ∈ dom dom 𝑔 ∧ 𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))) ↔ ((𝑥 ∈ dom dom 𝐺 ∧ 𝑦 ∈ dom dom 𝐺 ∧ 𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
17	16	albidv 1921	. . . . 5 ⊢ (𝑔 = 𝐺 → (∀𝑧((𝑥 ∈ dom dom 𝑔 ∧ 𝑦 ∈ dom dom 𝑔 ∧ 𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))) ↔ ∀𝑧((𝑥 ∈ dom dom 𝐺 ∧ 𝑦 ∈ dom dom 𝐺 ∧ 𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
18	17	2albidv 1924	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑥∀𝑦∀𝑧((𝑥 ∈ dom dom 𝑔 ∧ 𝑦 ∈ dom dom 𝑔 ∧ 𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ dom dom 𝐺 ∧ 𝑦 ∈ dom dom 𝐺 ∧ 𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
19		r3al 3174	. . . 4 ⊢ (∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ dom dom 𝑔 ∧ 𝑦 ∈ dom dom 𝑔 ∧ 𝑧 ∈ dom dom 𝑔) → ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))))
20		r3al 3174	. . . 4 ⊢ (∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ dom dom 𝐺 ∧ 𝑦 ∈ dom dom 𝐺 ∧ 𝑧 ∈ dom dom 𝐺) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
21	18, 19, 20	3bitr4g 314	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
22		isass.1	. . . . . 6 ⊢ 𝑋 = dom dom 𝐺
23	22	eqcomi 2745	. . . . 5 ⊢ dom dom 𝐺 = 𝑋
24	23	a1i 11	. . . 4 ⊢ (𝑔 = 𝐺 → dom dom 𝐺 = 𝑋)
25	24	raleqdv 3296	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
26	24, 25	raleqbidv 3316	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ dom dom 𝐺∀𝑦 ∈ dom dom 𝐺∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
27	24	raleqdv 3296	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
28	27	2ralbidv 3200	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
29	21, 26, 28	3bitrd 305	. 2 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
30		df-ass 38044	. 2 ⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
31	29, 30	elab2g 3635	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3051  dom cdm 5624  (class class class)co 7358  Asscass 38043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-dm 5634  df-iota 6448  df-fv 6500  df-ov 7361  df-ass 38044

This theorem is referenced by:  issmgrpOLD  38064