Theorem dmdprdsplitlem 20057

Description: Lemma for dmdprdsplit 20067. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
dmdprdsplitlem.0	⊢ 0 = (0g‘𝐺)
dmdprdsplitlem.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
dmdprdsplitlem.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dmdprdsplitlem.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dmdprdsplitlem.3	⊢ (𝜑 → 𝐴 ⊆ 𝐼)
dmdprdsplitlem.4	⊢ (𝜑 → 𝐹 ∈ 𝑊)
dmdprdsplitlem.5	⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆 ↾ 𝐴)))

Assertion

Ref	Expression
dmdprdsplitlem	⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

Distinct variable groups:   0 ,ℎ   ℎ,𝑖,𝐴   ℎ,𝐺,𝑖   ℎ,𝐼,𝑖   ℎ,𝐹   𝑆,ℎ,𝑖

Allowed substitution hints:   𝜑(ℎ,𝑖)   𝐹(𝑖)   𝑊(ℎ,𝑖)   𝑋(ℎ,𝑖)   0 (𝑖)

Proof of Theorem dmdprdsplitlem

Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmdprdsplitlem.5	. . . . 5 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆 ↾ 𝐴)))
2		dmdprdsplitlem.1	. . . . . . . 8 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
3		dmdprdsplitlem.2	. . . . . . . 8 ⊢ (𝜑 → dom 𝑆 = 𝐼)
4	2, 3	dprdf2 20027	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
5		dmdprdsplitlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
6	4, 5	fssresd 6775	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
7		fdm 6745	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐴) = 𝐴)
8		dmdprdsplitlem.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
9		eqid 2737	. . . . . . 7 ⊢ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 20024	. . . . . 6 ⊢ (dom (𝑆 ↾ 𝐴) = 𝐴 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆 ↾ 𝐴)) ↔ (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
11	6, 7, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd (𝑆 ↾ 𝐴)) ↔ (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
12	1, 11	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
13	12	simprd 495	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
14	13	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
15		simprr 773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
16	12	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
17	16	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐺dom DProd (𝑆 ↾ 𝐴))
18	6, 7	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴)
19	18	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → dom (𝑆 ↾ 𝐴) = 𝐴)
20		simprl 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 })
21		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
22	9, 17, 19, 20, 21	dprdff 20032	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑓:𝐴⟶(Base‘𝐺))
23	22	feqmptd 6977	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑓 = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛)))
24	5	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐴 ⊆ 𝐼)
25	24	resmptd 6058	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ↾ 𝐴) = (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )))
26		iftrue 4531	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝐴 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = (𝑓‘𝑛))
27	26	mpteq2ia 5245	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛))
28	25, 27	eqtrdi 2793	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ↾ 𝐴) = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛)))
29	23, 28	eqtr4d 2780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑓 = ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ↾ 𝐴))
30	29	oveq2d 7447	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐺 Σg 𝑓) = (𝐺 Σg ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ↾ 𝐴)))
31		eqid 2737	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
32	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐺dom DProd 𝑆)
33		dprdgrp 20025	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
34		grpmnd 18958	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
35	32, 33, 34	3syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐺 ∈ Mnd)
36	2, 3	dprddomcld 20021	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
37	36	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐼 ∈ V)
38		dmdprdsplitlem.w	. . . . . . . 8 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
39	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → dom 𝑆 = 𝐼)
40	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) → 𝐺dom DProd (𝑆 ↾ 𝐴))
41	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) → dom (𝑆 ↾ 𝐴) = 𝐴)
42		simplrl 777	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 })
43	9, 40, 41, 42	dprdfcl 20033	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 ∈ 𝐴) → (𝑓‘𝑛) ∈ ((𝑆 ↾ 𝐴)‘𝑛))
44		fvres 6925	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑛) = (𝑆‘𝑛))
45	44	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 ∈ 𝐴) → ((𝑆 ↾ 𝐴)‘𝑛) = (𝑆‘𝑛))
46	43, 45	eleqtrd 2843	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 ∈ 𝐴) → (𝑓‘𝑛) ∈ (𝑆‘𝑛))
47	4	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑆:𝐼⟶(SubGrp‘𝐺))
48	47	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) → (𝑆‘𝑛) ∈ (SubGrp‘𝐺))
49	8	subg0cl 19152	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑛) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑛))
50	48, 49	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) → 0 ∈ (𝑆‘𝑛))
51	50	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) ∧ ¬ 𝑛 ∈ 𝐴) → 0 ∈ (𝑆‘𝑛))
52	46, 51	ifclda 4561	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ 𝐼) → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) ∈ (𝑆‘𝑛))
53	36	mptexd 7244	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V)
54	53	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V)
55		funmpt 6604	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
56	55	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )))
57	9, 17, 19, 20	dprdffsupp 20034	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑓 finSupp 0 )
58		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
59		eldifn 4132	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 )) → ¬ 𝑛 ∈ (𝑓 supp 0 ))
60	59	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) → ¬ 𝑛 ∈ (𝑓 supp 0 ))
61	58, 60	eldifd 3962	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ (𝐴 ∖ (𝑓 supp 0 )))
62		ssidd 4007	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝑓 supp 0 ) ⊆ (𝑓 supp 0 ))
63	36, 5	ssexd 5324	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ V)
64	63	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐴 ∈ V)
65	8	fvexi 6920	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
66	65	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 0 ∈ V)
67	22, 62, 64, 66	suppssr 8220	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐴 ∖ (𝑓 supp 0 ))) → (𝑓‘𝑛) = 0 )
68	67	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) ∧ 𝑛 ∈ (𝐴 ∖ (𝑓 supp 0 ))) → (𝑓‘𝑛) = 0 )
69	61, 68	syldan 591	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) ∧ 𝑛 ∈ 𝐴) → (𝑓‘𝑛) = 0 )
70	69	ifeq1da 4557	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = if(𝑛 ∈ 𝐴, 0 , 0 ))
71		ifid 4566	. . . . . . . . . . . 12 ⊢ if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
72	70, 71	eqtrdi 2793	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = 0 )
73	72, 37	suppss2 8225	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) supp 0 ) ⊆ (𝑓 supp 0 ))
74		fsuppsssupp 9421	. . . . . . . . . 10 ⊢ ((((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V ∧ Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))) ∧ (𝑓 finSupp 0 ∧ ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) supp 0 ) ⊆ (𝑓 supp 0 ))) → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) finSupp 0 )
75	54, 56, 57, 73, 74	syl22anc 839	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) finSupp 0 )
76	38, 32, 39, 52, 75	dprdwd 20031	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ 𝑊)
77	38, 32, 39, 76, 21	dprdff 20032	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )):𝐼⟶(Base‘𝐺))
78	38, 32, 39, 76, 31	dprdfcntz 20035	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ran (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ⊆ ((Cntz‘𝐺)‘ran (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))))
79		eldifn 4132	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
80	79	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ 𝐴)) → ¬ 𝑛 ∈ 𝐴)
81	80	iffalsed 4536	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) ∧ 𝑛 ∈ (𝐼 ∖ 𝐴)) → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = 0 )
82	81, 37	suppss2 8225	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) supp 0 ) ⊆ 𝐴)
83	21, 8, 31, 35, 37, 77, 78, 82, 75	gsumzres 19927	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐺 Σg ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ↾ 𝐴)) = (𝐺 Σg (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))))
84	15, 30, 83	3eqtrd 2781	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))))
85		dmdprdsplitlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
86	85	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐹 ∈ 𝑊)
87	8, 38, 32, 39, 86, 76	dprdf11 20043	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ((𝐺 Σg 𝐹) = (𝐺 Σg (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))) ↔ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))))
88	84, 87	mpbid 232	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )))
89	88	fveq1d 6908	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐹‘𝑋) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))‘𝑋))
90		eldifi 4131	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → 𝑋 ∈ 𝐼)
91	90	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑋 ∈ 𝐼)
92		eleq1 2829	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
93		fveq2 6906	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑓‘𝑛) = (𝑓‘𝑋))
94	92, 93	ifbieq1d 4550	. . . . 5 ⊢ (𝑛 = 𝑋 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
95		eqid 2737	. . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
96		fvex 6919	. . . . . 6 ⊢ (𝑓‘𝑛) ∈ V
97	96, 65	ifex 4576	. . . . 5 ⊢ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) ∈ V
98	94, 95, 97	fvmpt3i 7021	. . . 4 ⊢ (𝑋 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))‘𝑋) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
99	91, 98	syl 17	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))‘𝑋) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
100		eldifn 4132	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑋 ∈ 𝐴)
101	100	ad2antlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ¬ 𝑋 ∈ 𝐴)
102	101	iffalsed 4536	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ) = 0 )
103	89, 99, 102	3eqtrd 2781	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐹‘𝑋) = 0 )
104	14, 103	rexlimddv 3161	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ifcif 4525   class class class wbr 5143   ↦ cmpt 5225  dom cdm 5685   ↾ cres 5687  Fun wfun 6555  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   supp csupp 8185  Xcixp 8937   finSupp cfsupp 9401  Basecbs 17247  0_gc0g 17484   Σ_g cgsu 17485  Mndcmnd 18747  Grpcgrp 18951  SubGrpcsubg 19138  Cntzccntz 19333   DProd cdprd 20013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-gim 19277  df-cntz 19335  df-oppg 19364  df-cmn 19800  df-dprd 20015

This theorem is referenced by:  dprddisj2  20059