Theorem dmdprdsplitlem 20132

Description: Lemma for dmdprdsplit 20142. (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 20102	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
5		dmdprdsplitlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
6	4, 5	fssresd 6749	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
7		fdm 6719	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐴) = 𝐴)
8		dmdprdsplitlem.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
9		eqid 2765	. . . . . . 7 ⊢ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 20099	. . . . . 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 19	. . . . 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 235	. . . 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 501	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
14	13	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
15		simprr 785	. . . . . 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 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
17	16	ad2antrr 739	. . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴)
19	18	ad2antrr 739	. . . . . . . . . 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 783	. . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
22	9, 17, 19, 20, 21	dprdff 20107	. . . . . . . . 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 6953	. . . . . . . 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 739	. . . . . . . . . 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 6044	. . . . . . . . 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 4495	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝐴 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = (𝑓‘𝑛))
27	26	mpteq2ia 5208	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛))
28	25, 27	eqtrdi 2816	. . . . . . . 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 2803	. . . . . . 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 7432	. . . . . 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 2765	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
32	2	ad2antrr 739	. . . . . . . 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 20100	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
34		grpmnd 19030	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
35	32, 33, 34	3syl 19	. . . . . . 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 20096	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
37	36	ad2antrr 739	. . . . . . 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 739	. . . . . . . 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 486	. . . . . . . . . . . 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 486	. . . . . . . . . . . 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 789	. . . . . . . . . . . 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 20108	. . . . . . . . . . 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 6904	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑛) = (𝑆‘𝑛))
45	44	adantl 487	. . . . . . . . . . 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 2867	. . . . . . . . . 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 739	. . . . . . . . . . . . 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 7083	. . . . . . . . . . . 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 19223	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑛) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑛))
50	48, 49	syl 18	. . . . . . . . . . 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 486	. . . . . . . . . 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 4525	. . . . . . . . 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 7226	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V)
54	53	ad2antrr 739	. . . . . . . . . 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 6578	. . . . . . . . . . 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 20109	. . . . . . . . . 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 490	. . . . . . . . . . . . . . 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 4086	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 )) → ¬ 𝑛 ∈ (𝑓 supp 0 ))
60	59	ad2antlr 740	. . . . . . . . . . . . . . 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 3917	. . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . 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 5297	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ V)
64	63	ad2antrr 739	. . . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . . . . . 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 8193	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 603	. . . . . . . . . . . . 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 4521	. . . . . . . . . . . 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 4530	. . . . . . . . . . . 12 ⊢ if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
72	70, 71	eqtrdi 2816	. . . . . . . . . . 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 8198	. . . . . . . . . 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 9344	. . . . . . . . . 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 852	. . . . . . . . 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 20106	. . . . . . . 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 20107	. . . . . . 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 20110	. . . . . . 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 4086	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
80	79	adantl 487	. . . . . . . . 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 4500	. . . . . . . 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 8198	. . . . . . 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 20002	. . . . . 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 2804	. . . . 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 739	. . . . . 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 20118	. . . . 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 235	. . . 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 6887	. . 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 4085	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → 𝑋 ∈ 𝐼)
91	90	ad2antlr 740	. . . 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 2853	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
93		fveq2 6885	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑓‘𝑛) = (𝑓‘𝑋))
94	92, 93	ifbieq1d 4514	. . . . 5 ⊢ (𝑛 = 𝑋 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
95		eqid 2765	. . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
96		fvex 6898	. . . . . 6 ⊢ (𝑓‘𝑛) ∈ V
97	96, 65	ifex 4540	. . . . 5 ⊢ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) ∈ V
98	94, 95, 97	fvmpt3i 6999	. . . 4 ⊢ (𝑋 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))‘𝑋) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
99	91, 98	syl 18	. . 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 4086	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑋 ∈ 𝐴)
101	100	ad2antlr 740	. . . 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 4500	. . 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 2804	. 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 3174	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  {crab 3418  Vcvv 3457   ∖ cdif 3903   ⊆ wss 3906  ifcif 4489   class class class wbr 5111   ↦ cmpt 5194  dom cdm 5663   ↾ cres 5665  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416   supp csupp 8158  Xcixp 8897   finSupp cfsupp 9324  Basecbs 17286  0_gc0g 17509   Σ_g cgsu 17510  Mndcmnd 18813  Grpcgrp 19023  SubGrpcsubg 19209  Cntzccntz 19408   DProd cdprd 20088

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  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  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-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  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-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9325  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-n0 12516  df-z 12603  df-uz 12874  df-fz 13547  df-fzo 13695  df-seq 14051  df-hash 14380  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-plusg 17340  df-0g 17511  df-gsum 17512  df-mre 17655  df-mrc 17656  df-acs 17658  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-mhm 18864  df-submnd 18865  df-grp 19026  df-minusg 19027  df-sbg 19028  df-mulg 19157  df-subg 19212  df-ghm 19307  df-gim 19352  df-cntz 19410  df-oppg 19439  df-cmn 19875  df-dprd 20090

This theorem is used by:  dprddisj2  20134