Theorem dmdprd 20094

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 3478	. . . . 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 7228	. . . . . . 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 3747	. . . . . 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 5897	. . . . . . . . . . 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 2800	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = 𝐼)
13	9, 12	feq12d 6697	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ:dom ℎ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
14	12	difeq1d 4080	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (dom ℎ ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
15	9	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑥) = (𝑆‘𝑥))
16	9	fveq1d 6887	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑦) = (𝑆‘𝑦))
17	16	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝑍‘(ℎ‘𝑦)) = (𝑍‘(𝑆‘𝑦)))
18	15, 17	sseq12d 3971	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ↔ (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))
19	14, 18	raleqbidv 3340	. . . . . . . . . . 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 6065	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ “ (dom ℎ ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
21	20	unieqd 4887	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ∪ (ℎ “ (dom ℎ ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
22	21	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
23	15, 22	ineq12d 4174	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
24	23	eqeq1d 2767	. . . . . . . . . . 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 3340	. . . . . . . . 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 3789	. . . . . 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 5112	. . 3 ⊢ (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35		fvex 6898	. . . . . . . . . . 11 ⊢ (𝑠‘𝑥) ∈ V
36	35	rgenw 3085	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V
37		ixpexg 8922	. . . . . . . . . 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 7227	. . . . . . . 8 ⊢ (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
40	39	rnex 7909	. . . . . . 7 ⊢ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
41	40	rgen2w 3086	. . . . . 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 20091	. . . . . . 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 8066	. . . . . 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 6721	. . . 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 2857	. . 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 6885	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
48	47	feq3d 6694	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ:dom ℎ⟶(SubGrp‘𝑔) ↔ ℎ:dom ℎ⟶(SubGrp‘𝐺)))
49		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50		dmdprd.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (Cntz‘𝐺)
51	49, 50	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
52	51	fveq1d 6887	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(ℎ‘𝑦)) = (𝑍‘(ℎ‘𝑦)))
53	52	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ↔ (ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦))))
54	53	ralbidv 3190	. . . . . . . 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 6889	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56		dmdprd.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	55, 56	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
58	57	fveq1d 6887	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))))
59	58	ineq2d 4173	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))))
60		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
61		dmdprd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺)
62	60, 61	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
63	62	sneqd 4603	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(0g‘𝑔)} = { 0 })
64	59, 63	eqeq12d 2781	. . . . . . . 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 3190	. . . . . 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 2831	. . . 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 5826	. . 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 2146  {cab 2743  ∀wral 3081  {crab 3418  Vcvv 3457  [wsbc 3746   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ⟨cop 4597  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  dom cdm 5663  ran crn 5664   “ cima 5666  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  Xcixp 8897   finSupp cfsupp 9324  0_gc0g 17510   Σ_g cgsu 17511  mrClscmrc 17653  Grpcgrp 19024  SubGrpcsubg 19210  Cntzccntz 19409   DProd cdprd 20089

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-ixp 8898  df-dprd 20091

This theorem is used by:  dmdprdd  20095  dprdgrp  20101  dprdf  20102  dprdcntz  20104  dprddisj  20105  dprdres  20124  subgdmdprd  20130