Theorem isass 37847

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 5870	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
2	1	dmeqd 5872	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → dom dom 𝑔 = dom dom 𝐺)
3	2	eleq2d 2815	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ dom dom 𝑔 ↔ 𝑥 ∈ dom dom 𝐺))
4	2	eleq2d 2815	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑦 ∈ dom dom 𝑔 ↔ 𝑦 ∈ dom dom 𝐺))
5	2	eleq2d 2815	. . . . . . . 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 7396	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
8	7	oveq1d 7405	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝑔𝑧))
9		oveq 7396	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((𝑥𝐺𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
10	8, 9	eqtrd 2765	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦)𝑔𝑧) = ((𝑥𝐺𝑦)𝐺𝑧))
11		oveq 7396	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑧) = (𝑦𝐺𝑧))
12	11	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝑔(𝑦𝐺𝑧)))
13		oveq 7396	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝐺𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
14	12, 13	eqtrd 2765	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥𝑔(𝑦𝑔𝑧)) = (𝑥𝐺(𝑦𝐺𝑧)))
15	10, 14	eqeq12d 2746	. . . . . . 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 1920	. . . . 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 1923	. . . 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 3176	. . . 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 3176	. . . 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 2739	. . . . 5 ⊢ dom dom 𝐺 = 𝑋
24	23	a1i 11	. . . 4 ⊢ (𝑔 = 𝐺 → dom dom 𝐺 = 𝑋)
25	24	raleqdv 3301	. . . 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 3321	. . 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 3301	. . . 4 ⊢ (𝑔 = 𝐺 → (∀𝑧 ∈ dom dom 𝐺((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) ↔ ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
28	27	2ralbidv 3202	. . 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 37844	. 2 ⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
31	29, 30	elab2g 3650	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3045  dom cdm 5641  (class class class)co 7390  Asscass 37843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-dm 5651  df-iota 6467  df-fv 6522  df-ov 7393  df-ass 37844

This theorem is referenced by:  issmgrpOLD  37864