Theorem dmdprd 20018

Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dmdprd.z	⊢ 𝑍 = (Cntz‘𝐺)
dmdprd.0	⊢ 0 = (0g‘𝐺)
dmdprd.k	⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))

Assertion

Ref	Expression
dmdprd	⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝐼,𝑦   𝑥,𝑆,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝐾(𝑥,𝑦)   0 (𝑥,𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem dmdprd

Dummy variables 𝑔 ℎ 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3501	. . . . 5 ⊢ (𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))} → 𝑆 ∈ V)
2	1	a1i 11	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))} → 𝑆 ∈ V))
3		fex 7246	. . . . . . 7 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼 ∈ 𝑉) → 𝑆 ∈ V)
4	3	expcom 413	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
5	4	adantr 480	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
6	5	adantrd 491	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) → 𝑆 ∈ V))
7		df-sbc 3789	. . . . . 6 ⊢ ([𝑆 / ℎ](ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 })) ↔ 𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))})
8		simpr 484	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9		simpr 484	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ℎ = 𝑆)
10	9	dmeqd 5916	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = dom 𝑆)
11		simplr 769	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom 𝑆 = 𝐼)
12	10, 11	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = 𝐼)
13	9, 12	feq12d 6724	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ:dom ℎ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
14	12	difeq1d 4125	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (dom ℎ ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
15	9	fveq1d 6908	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑥) = (𝑆‘𝑥))
16	9	fveq1d 6908	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑦) = (𝑆‘𝑦))
17	16	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝑍‘(ℎ‘𝑦)) = (𝑍‘(𝑆‘𝑦)))
18	15, 17	sseq12d 4017	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ↔ (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))
19	14, 18	raleqbidv 3346	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ↔ ∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))
20	9, 14	imaeq12d 6079	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ “ (dom ℎ ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
21	20	unieqd 4920	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ∪ (ℎ “ (dom ℎ ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
22	21	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
23	15, 22	ineq12d 4221	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
24	23	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 } ↔ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))
25	19, 24	anbi12d 632	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }) ↔ (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
26	12, 25	raleqbidv 3346	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }) ↔ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))
27	13, 26	anbi12d 632	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 })) ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
28	27	adantlr 715	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) ∧ ℎ = 𝑆) → ((ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 })) ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
29	8, 28	sbcied 3832	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → ([𝑆 / ℎ](ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 })) ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
30	7, 29	bitr3id 285	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → (𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))} ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
31	30	ex 412	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆 ∈ V → (𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))} ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))))
32	2, 6, 31	pm5.21ndd 379	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))} ↔ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
33	32	anbi2d 630	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → ((𝐺 ∈ Grp ∧ 𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))}) ↔ (𝐺 ∈ Grp ∧ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })))))
34		df-br 5144	. . 3 ⊢ (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35		fvex 6919	. . . . . . . . . . 11 ⊢ (𝑠‘𝑥) ∈ V
36	35	rgenw 3065	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V
37		ixpexg 8962	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V → X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V)
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V
39	38	mptrabex 7245	. . . . . . . 8 ⊢ (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
40	39	rnex 7932	. . . . . . 7 ⊢ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
41	40	rgen2w 3066	. . . . . 6 ⊢ ∀𝑔 ∈ Grp ∀𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))}ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
42		df-dprd 20015	. . . . . . 7 ⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
43	42	fmpox 8092	. . . . . 6 ⊢ (∀𝑔 ∈ Grp ∀𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))}ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V ↔ DProd :∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))})⟶V)
44	41, 43	mpbi 230	. . . . 5 ⊢ DProd :∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))})⟶V
45	44	fdmi 6747	. . . 4 ⊢ dom DProd = ∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))})
46	45	eleq2i 2833	. . 3 ⊢ (⟨𝐺, 𝑆⟩ ∈ dom DProd ↔ ⟨𝐺, 𝑆⟩ ∈ ∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))}))
47		fveq2 6906	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
48	47	feq3d 6723	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ:dom ℎ⟶(SubGrp‘𝑔) ↔ ℎ:dom ℎ⟶(SubGrp‘𝐺)))
49		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50		dmdprd.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (Cntz‘𝐺)
51	49, 50	eqtr4di 2795	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
52	51	fveq1d 6908	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(ℎ‘𝑦)) = (𝑍‘(ℎ‘𝑦)))
53	52	sseq2d 4016	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ↔ (ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦))))
54	53	ralbidv 3178	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ↔ ∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦))))
55	47	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56		dmdprd.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	55, 56	eqtr4di 2795	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
58	57	fveq1d 6908	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))))
59	58	ineq2d 4220	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))))
60		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
61		dmdprd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺)
62	60, 61	eqtr4di 2795	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
63	62	sneqd 4638	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(0g‘𝑔)} = { 0 })
64	59, 63	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)} ↔ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))
65	54, 64	anbi12d 632	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}) ↔ (∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 })))
66	65	ralbidv 3178	. . . . . 6 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}) ↔ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 })))
67	48, 66	anbi12d 632	. . . . 5 ⊢ (𝑔 = 𝐺 → ((ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)})) ↔ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))))
68	67	abbidv 2808	. . . 4 ⊢ (𝑔 = 𝐺 → {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))} = {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))})
69	68	opeliunxp2 5849	. . 3 ⊢ (⟨𝐺, 𝑆⟩ ∈ ∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))}) ↔ (𝐺 ∈ Grp ∧ 𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))}))
70	34, 46, 69	3bitri 297	. 2 ⊢ (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = { 0 }))}))
71		3anass 1095	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })) ↔ (𝐺 ∈ Grp ∧ (𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))
72	33, 70, 71	3bitr4g 314	1 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:𝐼⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ (𝐼 ∖ {𝑥})(𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 }))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  {crab 3436  Vcvv 3480  [wsbc 3788   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  {csn 4626  ⟨cop 4632  ∪ cuni 4907  ∪ ciun 4991   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686   “ cima 5688  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Xcixp 8937   finSupp cfsupp 9401  0_gc0g 17484   Σ_g cgsu 17485  mrClscmrc 17626  Grpcgrp 18951  SubGrpcsubg 19138  Cntzccntz 19333   DProd cdprd 20013

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-ixp 8938  df-dprd 20015

This theorem is referenced by:  dmdprdd  20019  dprdgrp  20025  dprdf  20026  dprdcntz  20028  dprddisj  20029  dprdres  20048  subgdmdprd  20054