Theorem dmdprd 20127

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 3471	. . . . 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 7225	. . . . . . 7 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼 ∈ 𝑉) → 𝑆 ∈ V)
4	3	expcom 419	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
5	4	adantr 486	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
6	5	adantrd 497	. . . 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 3740	. . . . . 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 490	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9		simpr 490	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ℎ = 𝑆)
10	9	dmeqd 5889	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = dom 𝑆)
11		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom 𝑆 = 𝐼)
12	10, 11	eqtrd 2795	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = 𝐼)
13	9, 12	feq12d 6690	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ:dom ℎ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
14	12	difeq1d 4073	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (dom ℎ ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
15	9	fveq1d 6880	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑥) = (𝑆‘𝑥))
16	9	fveq1d 6880	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑦) = (𝑆‘𝑦))
17	16	fveq2d 6882	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝑍‘(ℎ‘𝑦)) = (𝑍‘(𝑆‘𝑦)))
18	15, 17	sseq12d 3964	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ↔ (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))
19	14, 18	raleqbidv 3334	. . . . . . . . . . 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 6057	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ “ (dom ℎ ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
21	20	unieqd 4880	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ∪ (ℎ “ (dom ℎ ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
22	21	fveq2d 6882	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
23	15, 22	ineq12d 4167	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
24	23	eqeq1d 2762	. . . . . . . . . . 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 644	. . . . . . . . . 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 3334	. . . . . . . . 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 644	. . . . . . . 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 728	. . . . . . 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 3782	. . . . . 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 288	. . . . 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 418	. . . 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 382	. . 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 642	. 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 5104	. . 3 ⊢ (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35		fvex 6891	. . . . . . . . . . 11 ⊢ (𝑠‘𝑥) ∈ V
36	35	rgenw 3080	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V
37		ixpexg 8929	. . . . . . . . . 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 7224	. . . . . . . 8 ⊢ (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
40	39	rnex 7907	. . . . . . 7 ⊢ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
41	40	rgen2w 3081	. . . . . 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 20124	. . . . . . 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 8064	. . . . . 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 233	. . . . 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 6714	. . . 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 2852	. . 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 6878	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
48	47	feq3d 6687	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ:dom ℎ⟶(SubGrp‘𝑔) ↔ ℎ:dom ℎ⟶(SubGrp‘𝐺)))
49		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50		dmdprd.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (Cntz‘𝐺)
51	49, 50	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
52	51	fveq1d 6880	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(ℎ‘𝑦)) = (𝑍‘(ℎ‘𝑦)))
53	52	sseq2d 3963	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ↔ (ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦))))
54	53	ralbidv 3185	. . . . . . . 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 6882	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56		dmdprd.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	55, 56	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
58	57	fveq1d 6880	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))))
59	58	ineq2d 4166	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))))
60		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
61		dmdprd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺)
62	60, 61	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
63	62	sneqd 4596	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(0g‘𝑔)} = { 0 })
64	59, 63	eqeq12d 2776	. . . . . . . 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 644	. . . . . . 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 3185	. . . . . 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 644	. . . . 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 2826	. . . 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 5818	. . 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 300	. 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 1111	. 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 317	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 }))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412  Vcvv 3450  [wsbc 3739   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  dom cdm 5655  ran crn 5656   “ cima 5658  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Xcixp 8904   finSupp cfsupp 9331  0_gc0g 17524   Σ_g cgsu 17525  mrClscmrc 17667  Grpcgrp 19057  SubGrpcsubg 19243  Cntzccntz 19442   DProd cdprd 20122

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-ixp 8905  df-dprd 20124

This theorem is used by:  dmdprdd  20128  dprdgrp  20134  dprdf  20135  dprdcntz  20137  dprddisj  20138  dprdres  20157  subgdmdprd  20163