Theorem dprdval 18225

Description: The value of the internal direct product operation, which is a function mapping the (infinite, but finitely supported) cartesian product of subgroups (which mutually commute and have trivial intersections) to its (group) sum . (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dprdval.0	⊢ 0 = (0g‘𝐺)
dprdval.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }

Assertion

Ref	Expression
dprdval	⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))

Distinct variable groups:   𝑓,ℎ,𝑖,𝐼   𝑆,𝑓,ℎ,𝑖   𝑓,𝐺,ℎ,𝑖

Allowed substitution hints:   𝑊(𝑓,ℎ,𝑖)   0 (𝑓,ℎ,𝑖)

Proof of Theorem dprdval

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

Step	Hyp	Ref	Expression
1		simpl 472	. 2 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2		reldmdprd 18219	. . . . . 6 ⊢ Rel dom DProd
3	2	brrelex2i 5083	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
4	3	adantr 480	. . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
5	2	brrelexi 5082	. . . . . 6 ⊢ (𝐺dom DProd 𝑠 → 𝐺 ∈ V)
6		breq1 4586	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔dom DProd 𝑠 ↔ 𝐺dom DProd 𝑠))
7		oveq1 6556	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
9		dprdval.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
10	8, 9	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
11	10	breq2d 4595	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (ℎ finSupp (0g‘𝑔) ↔ ℎ finSupp 0 ))
12	11	rabbidv 3164	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} = {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 })
13		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
14	12, 13	mpteq12dv 4663	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) = (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
15	14	rneqd 5274	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
16	7, 15	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑔 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ↔ (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
17	6, 16	imbi12d 333	. . . . . . 7 ⊢ (𝑔 = 𝐺 → ((𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓))) ↔ (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)))))
18		df-br 4584	. . . . . . . . 9 ⊢ (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠‘𝑖) ∈ V
20	19	rgenw 2908	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∈ V
21		ixpexg 7818	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∈ V → X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∈ V)
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∈ V
23	22	mptrabex 6392	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
24	23	rnex 6992	. . . . . . . . . . . . 13 ⊢ ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
25	24	rgen2w 2909	. . . . . . . . . . . 12 ⊢ ∀𝑔 ∈ Grp ∀𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
26		df-dprd 18217	. . . . . . . . . . . . 13 ⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
27	26	fmpt2x 7125	. . . . . . . . . . . 12 ⊢ (∀𝑔 ∈ Grp ∀𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V ↔ DProd :∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))})⟶V)
28	25, 27	mpbi 219	. . . . . . . . . . 11 ⊢ DProd :∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))})⟶V
29	28	fdmi 5965	. . . . . . . . . 10 ⊢ dom DProd = ∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))})
30	29	eleq2i 2680	. . . . . . . . 9 ⊢ (⟨𝑔, 𝑠⟩ ∈ dom DProd ↔ ⟨𝑔, 𝑠⟩ ∈ ∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}))
31		opeliunxp 5093	. . . . . . . . 9 ⊢ (⟨𝑔, 𝑠⟩ ∈ ∪ 𝑔 ∈ Grp ({𝑔} × {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}) ↔ (𝑔 ∈ Grp ∧ 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}))
32	18, 30, 31	3bitri 285	. . . . . . . 8 ⊢ (𝑔dom DProd 𝑠 ↔ (𝑔 ∈ Grp ∧ 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}))
33	26	ovmpt4g 6681	. . . . . . . . 9 ⊢ ((𝑔 ∈ Grp ∧ 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))} ∧ ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V) → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
34	24, 33	mp3an3 1405	. . . . . . . 8 ⊢ ((𝑔 ∈ Grp ∧ 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑖 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑖})(ℎ‘𝑖) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑖) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑖})))) = {(0g‘𝑔)}))}) → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
35	32, 34	sylbi 206	. . . . . . 7 ⊢ (𝑔dom DProd 𝑠 → (𝑔 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
36	17, 35	vtoclg 3239	. . . . . 6 ⊢ (𝐺 ∈ V → (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
37	5, 36	mpcom 37	. . . . 5 ⊢ (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
38	37	sbcth 3417	. . . 4 ⊢ (𝑆 ∈ V → [𝑆 / 𝑠](𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
39	4, 38	syl 17	. . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → [𝑆 / 𝑠](𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓))))
40		simpr 476	. . . . . 6 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
41	40	breq2d 4595	. . . . 5 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑆))
42	40	oveq2d 6565	. . . . . 6 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
43	40	dmeqd 5248	. . . . . . . . . . . . 13 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = dom 𝑆)
44		simplr 788	. . . . . . . . . . . . 13 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑆 = 𝐼)
45	43, 44	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = 𝐼)
46	45	ixpeq1d 7806	. . . . . . . . . . 11 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠‘𝑖) = X𝑖 ∈ 𝐼 (𝑠‘𝑖))
47	40	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠‘𝑖) = (𝑆‘𝑖))
48	47	ixpeq2dv 7810	. . . . . . . . . . 11 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ 𝐼 (𝑠‘𝑖) = X𝑖 ∈ 𝐼 (𝑆‘𝑖))
49	46, 48	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠‘𝑖) = X𝑖 ∈ 𝐼 (𝑆‘𝑖))
50	49	rabeqdv 3167	. . . . . . . . 9 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 })
51		dprdval.w	. . . . . . . . 9 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
52	50, 51	syl6eqr 2662	. . . . . . . 8 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } = 𝑊)
53		eqidd 2611	. . . . . . . 8 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
54	52, 53	mpteq12dv 4663	. . . . . . 7 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))
55	54	rneqd 5274	. . . . . 6 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))
56	42, 55	eqeq12d 2625	. . . . 5 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → ((𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)) ↔ (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))))
57	41, 56	imbi12d 333	. . . 4 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → ((𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓))) ↔ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))))
58	4, 57	sbcied 3439	. . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → ([𝑆 / 𝑠](𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓))) ↔ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))))
59	39, 58	mpbid 221	. 2 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓))))
60	1, 59	mpd 15	1 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ 𝑊 ↦ (𝐺 Σg 𝑓)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Xcixp 7794   finSupp cfsupp 8158  0_gc0g 15923   Σ_g cgsu 15924  mrClscmrc 16066  Grpcgrp 17245  SubGrpcsubg 17411  Cntzccntz 17571   DProd cdprd 18215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ixp 7795  df-dprd 18217

This theorem is referenced by:  eldprd  18226  dprdlub  18248