Theorem dmdprdsplitlem 18436

Description: Lemma for dmdprdsplit 18446. (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 18406	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
5		dmdprdsplitlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
6	4, 5	fssresd 6071	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
7		fdm 6051	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐴) = 𝐴)
8		dmdprdsplitlem.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
9		eqid 2622	. . . . . . 7 ⊢ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 18403	. . . . . 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 222	. . . 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 479	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
14	13	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
15		simprr 796	. . . . . 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 475	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
17	16	ad2antrr 762	. . . . . . . . . 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 762	. . . . . . . . . 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 794	. . . . . . . . . 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 2622	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
22	9, 17, 19, 20, 21	dprdff 18411	. . . . . . . . 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 6249	. . . . . . . 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 762	. . . . . . . . . 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 5452	. . . . . . . . 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 4092	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝐴 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = (𝑓‘𝑛))
27	26	mpteq2ia 4740	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛))
28	25, 27	syl6eq 2672	. . . . . . . 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 2659	. . . . . . 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 6666	. . . . . 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 2622	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
32	2	ad2antrr 762	. . . . . . . 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 18404	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
34		grpmnd 17429	. . . . . . . 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 18400	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
37	36	ad2antrr 762	. . . . . . 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 762	. . . . . . . 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 481	. . . . . . . . . . . 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 481	. . . . . . . . . . . 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 800	. . . . . . . . . . . 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 18412	. . . . . . . . . . 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 6207	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑛) = (𝑆‘𝑛))
45	44	adantl 482	. . . . . . . . . . 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 2703	. . . . . . . . . 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 762	. . . . . . . . . . . . 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	ffvelrnda 6359	. . . . . . . . . . . 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 17602	. . . . . . . . . . . 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 481	. . . . . . . . . 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 4120	. . . . . . . . 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		mptexg 6484	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ V → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V)
54	36, 53	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V)
55	54	ad2antrr 762	. . . . . . . . . 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)
56		funmpt 5926	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
57	56	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 )))
58	9, 17, 19, 20	dprdffsupp 18413	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑓 finSupp 0 )
59		simpr 477	. . . . . . . . . . . . . . 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)
60		eldifn 3733	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 )) → ¬ 𝑛 ∈ (𝑓 supp 0 ))
61	60	ad2antlr 763	. . . . . . . . . . . . . . 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 ))
62	59, 61	eldifd 3585	. . . . . . . . . . . . . 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 )))
63		ssid 3624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 supp 0 ) ⊆ (𝑓 supp 0 )
64	63	a1i 11	. . . . . . . . . . . . . . . 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 ))
65	36, 5	ssexd 4805	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ V)
66	65	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐴 ∈ V)
67		fvex 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝐺) ∈ V
68	8, 67	eqeltri 2697	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
69	68	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)
70	22, 64, 66, 69	suppssr 7326	. . . . . . . . . . . . . . 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 )
71	70	adantlr 751	. . . . . . . . . . . . . 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 )
72	62, 71	syldan 487	. . . . . . . . . . . . 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 )
73	72	ifeq1da 4116	. . . . . . . . . . . 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 ))
74		ifid 4125	. . . . . . . . . . . 12 ⊢ if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
75	73, 74	syl6eq 2672	. . . . . . . . . . 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 )
76	75, 37	suppss2 7329	. . . . . . . . . 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 ))
77		fsuppsssupp 8291	. . . . . . . . . 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 )
78	55, 57, 58, 76, 77	syl22anc 1327	. . . . . . . . 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 )
79	38, 32, 39, 52, 78	dprdwd 18410	. . . . . . . 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)
80	38, 32, 39, 79, 21	dprdff 18411	. . . . . . 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))
81	38, 32, 39, 79, 31	dprdfcntz 18414	. . . . . . 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 ))))
82		eldifn 3733	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
83	82	adantl 482	. . . . . . . . 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)
84	83	iffalsed 4097	. . . . . . . 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 )
85	84, 37	suppss2 7329	. . . . . . 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)
86	21, 8, 31, 35, 37, 80, 81, 85, 78	gsumzres 18310	. . . . . 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 ))))
87	15, 30, 86	3eqtrd 2660	. . . . 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 ))))
88		dmdprdsplitlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
89	88	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝐹 ∈ 𝑊)
90	8, 38, 32, 39, 89, 79	dprdf11 18422	. . . . 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 ))))
91	87, 90	mpbid 222	. . . 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 )))
92	91	fveq1d 6193	. . 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))
93		eldifi 3732	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → 𝑋 ∈ 𝐼)
94	93	ad2antlr 763	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → 𝑋 ∈ 𝐼)
95		eleq1 2689	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
96		fveq2 6191	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑓‘𝑛) = (𝑓‘𝑋))
97	95, 96	ifbieq1d 4109	. . . . 5 ⊢ (𝑛 = 𝑋 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
98		eqid 2622	. . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
99		fvex 6201	. . . . . 6 ⊢ (𝑓‘𝑛) ∈ V
100	99, 68	ifex 4156	. . . . 5 ⊢ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) ∈ V
101	97, 98, 100	fvmpt3i 6287	. . . 4 ⊢ (𝑋 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))‘𝑋) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
102	94, 101	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 ))
103		eldifn 3733	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑋 ∈ 𝐴)
104	103	ad2antlr 763	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → ¬ 𝑋 ∈ 𝐴)
105	104	iffalsed 4097	. . 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 )
106	92, 102, 105	3eqtrd 2660	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) ∧ (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))) → (𝐹‘𝑋) = 0 )
107	14, 106	rexlimddv 3035	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114   ↾ cres 5116  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   supp csupp 7295  Xcixp 7908   finSupp cfsupp 8275  Basecbs 15857  0_gc0g 16100   Σ_g cgsu 16101  Mndcmnd 17294  Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-cmn 18195  df-dprd 18394

This theorem is referenced by:  dprddisj2  18438