Theorem dmdprd 19966

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 3452	. . . . 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 7170	. . . . . . 7 ⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐼 ∈ 𝑉) → 𝑆 ∈ V)
4	3	expcom 414	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
5	4	adantr 481	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) → (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 ∈ V))
6	5	adantrd 492	. . . 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 3724	. . . . . 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 485	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
9		simpr 485	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ℎ = 𝑆)
10	9	dmeqd 5847	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = dom 𝑆)
11		simplr 774	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom 𝑆 = 𝐼)
12	10, 11	eqtrd 2774	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → dom ℎ = 𝐼)
13	9, 12	feq12d 6643	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ:dom ℎ⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
14	12	difeq1d 4056	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (dom ℎ ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
15	9	fveq1d 6829	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑥) = (𝑆‘𝑥))
16	9	fveq1d 6829	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ‘𝑦) = (𝑆‘𝑦))
17	16	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝑍‘(ℎ‘𝑦)) = (𝑍‘(𝑆‘𝑦)))
18	15, 17	sseq12d 3948	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦)) ↔ (𝑆‘𝑥) ⊆ (𝑍‘(𝑆‘𝑦))))
19	14, 18	raleqbidv 3313	. . . . . . . . . . 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 6013	. . . . . . . . . . . . . . 15 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (ℎ “ (dom ℎ ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {𝑥})))
21	20	unieqd 4851	. . . . . . . . . . . . . 14 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ∪ (ℎ “ (dom ℎ ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
22	21	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
23	15, 22	ineq12d 4150	. . . . . . . . . . . 12 ⊢ (((𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼) ∧ ℎ = 𝑆) → ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
24	23	eqeq1d 2741	. . . . . . . . . . 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 638	. . . . . . . . . 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 3313	. . . . . . . . 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 638	. . . . . . . 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 721	. . . . . . 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 3766	. . . . . 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 286	. . . . 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 413	. . . 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 380	. . 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 636	. 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 5073	. . 3 ⊢ (𝐺dom DProd 𝑆 ↔ ⟨𝐺, 𝑆⟩ ∈ dom DProd )
35		fvex 6840	. . . . . . . . . . 11 ⊢ (𝑠‘𝑥) ∈ V
36	35	rgenw 3057	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∈ V
37		ixpexg 8860	. . . . . . . . . 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 7169	. . . . . . . 8 ⊢ (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
40	39	rnex 7850	. . . . . . 7 ⊢ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
41	40	rgen2w 3058	. . . . . 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 19963	. . . . . . 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 8009	. . . . . 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 231	. . . . 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 6666	. . . 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 2831	. . 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 6827	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
48	47	feq3d 6640	. . . . . 6 ⊢ (𝑔 = 𝐺 → (ℎ:dom ℎ⟶(SubGrp‘𝑔) ↔ ℎ:dom ℎ⟶(SubGrp‘𝐺)))
49		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = (Cntz‘𝐺))
50		dmdprd.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (Cntz‘𝐺)
51	49, 50	eqtr4di 2792	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Cntz‘𝑔) = 𝑍)
52	51	fveq1d 6829	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Cntz‘𝑔)‘(ℎ‘𝑦)) = (𝑍‘(ℎ‘𝑦)))
53	52	sseq2d 3947	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ↔ (ℎ‘𝑥) ⊆ (𝑍‘(ℎ‘𝑦))))
54	53	ralbidv 3162	. . . . . . . 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 6831	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = (mrCls‘(SubGrp‘𝐺)))
56		dmdprd.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))
57	55, 56	eqtr4di 2792	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (mrCls‘(SubGrp‘𝑔)) = 𝐾)
58	57	fveq1d 6829	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))) = (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥}))))
59	58	ineq2d 4149	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = ((ℎ‘𝑥) ∩ (𝐾‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))))
60		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
61		dmdprd.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝐺)
62	60, 61	eqtr4di 2792	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
63	62	sneqd 4567	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → {(0g‘𝑔)} = { 0 })
64	59, 63	eqeq12d 2755	. . . . . . . 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 638	. . . . . . 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 3162	. . . . . 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 638	. . . . 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 2805	. . . 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 5780	. . 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 298	. 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 1100	. 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 315	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  {crab 3391  Vcvv 3431  [wsbc 3723   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  {csn 4555  ⟨cop 4561  ∪ cuni 4838  ∪ ciun 4921   class class class wbr 5072   ↦ cmpt 5153   × cxp 5616  dom cdm 5618  ran crn 5619   “ cima 5621  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Xcixp 8835   finSupp cfsupp 9264  0_gc0g 17393   Σ_g cgsu 17394  mrClscmrc 17536  Grpcgrp 18900  SubGrpcsubg 19087  Cntzccntz 19281   DProd cdprd 19961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-ixp 8836  df-dprd 19963

This theorem is referenced by:  dmdprdd  19967  dprdgrp  19973  dprdf  19974  dprdcntz  19976  dprddisj  19977  dprdres  19996  subgdmdprd  20002