Theorem dprdval 19872

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 483	. 2 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝐺dom DProd 𝑆)
2		reldmdprd 19866	. . . . . 6 ⊢ Rel dom DProd
3	2	brrelex2i 5733	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
4	3	adantr 481	. . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → 𝑆 ∈ V)
5	2	brrelex1i 5732	. . . . . 6 ⊢ (𝐺dom DProd 𝑠 → 𝐺 ∈ V)
6		breq1 5151	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔dom DProd 𝑠 ↔ 𝐺dom DProd 𝑠))
7		oveq1 7415	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑔 DProd 𝑠) = (𝐺 DProd 𝑠))
8		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = (0g‘𝐺))
9		dprdval.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
10	8, 9	eqtr4di 2790	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (0g‘𝑔) = 0 )
11	10	breq2d 5160	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (ℎ finSupp (0g‘𝑔) ↔ ℎ finSupp 0 ))
12	11	rabbidv 3440	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} = {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 })
13		oveq1 7415	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑔 Σg 𝑓) = (𝐺 Σg 𝑓))
14	12, 13	mpteq12dv 5239	. . . . . . . . . 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 5937	. . . . . . . . 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 2748	. . . . . . . 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 344	. . . . . . 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 5149	. . . . . . . . 9 ⊢ (𝑔dom DProd 𝑠 ↔ ⟨𝑔, 𝑠⟩ ∈ dom DProd )
19		fvex 6904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠‘𝑖) ∈ V
20	19	rgenw 3065	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∈ V
21		ixpexg 8915	. . . . . . . . . . . . . . . 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 7226	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
24	23	rnex 7902	. . . . . . . . . . . . 13 ⊢ ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)) ∈ V
25	24	rgen2w 3066	. . . . . . . . . . . 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 19864	. . . . . . . . . . . . 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	fmpox 8052	. . . . . . . . . . . 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 229	. . . . . . . . . . 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 6729	. . . . . . . . . 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 2825	. . . . . . . . 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 5743	. . . . . . . . 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 296	. . . . . . . 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 7554	. . . . . . . . 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 1450	. . . . . . . 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 216	. . . . . . 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 3556	. . . . . 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 38	. . . . 5 ⊢ (𝐺dom DProd 𝑠 → (𝐺 DProd 𝑠) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } ↦ (𝐺 Σg 𝑓)))
38	37	sbcth 3792	. . . 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 485	. . . . . 6 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
41	40	breq2d 5160	. . . . 5 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑆))
42	40	oveq2d 7424	. . . . . 6 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑆))
43	40	dmeqd 5905	. . . . . . . . . . . . 13 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = dom 𝑆)
44		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑆 = 𝐼)
45	43, 44	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → dom 𝑠 = 𝐼)
46	45	ixpeq1d 8902	. . . . . . . . . . 11 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠‘𝑖) = X𝑖 ∈ 𝐼 (𝑠‘𝑖))
47	40	fveq1d 6893	. . . . . . . . . . . 12 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝑠‘𝑖) = (𝑆‘𝑖))
48	47	ixpeq2dv 8906	. . . . . . . . . . 11 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ 𝐼 (𝑠‘𝑖) = X𝑖 ∈ 𝐼 (𝑆‘𝑖))
49	46, 48	eqtrd 2772	. . . . . . . . . 10 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → X𝑖 ∈ dom 𝑠(𝑠‘𝑖) = X𝑖 ∈ 𝐼 (𝑆‘𝑖))
50	49	rabeqdv 3447	. . . . . . . . 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	eqtr4di 2790	. . . . . . . 8 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → {ℎ ∈ X𝑖 ∈ dom 𝑠(𝑠‘𝑖) ∣ ℎ finSupp 0 } = 𝑊)
53		eqidd 2733	. . . . . . . 8 ⊢ (((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) ∧ 𝑠 = 𝑆) → (𝐺 Σg 𝑓) = (𝐺 Σg 𝑓))
54	52, 53	mpteq12dv 5239	. . . . . . 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 5937	. . . . . 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 2748	. . . . 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 344	. . . 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 3822	. . 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 231	. 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  {crab 3432  Vcvv 3474  [wsbc 3777   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948  {csn 4628  ⟨cop 4634  ∪ cuni 4908  ∪ ciun 4997   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674  dom cdm 5676  ran crn 5677   “ cima 5679  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408  Xcixp 8890   finSupp cfsupp 9360  0_gc0g 17384   Σ_g cgsu 17385  mrClscmrc 17526  Grpcgrp 18818  SubGrpcsubg 18999  Cntzccntz 19178   DProd cdprd 19862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-ixp 8891  df-dprd 19864

This theorem is referenced by:  eldprd  19873  dprdlub  19895