Theorem dmdprdsplitlem 18259

Description: Lemma for dmdprdsplit 18269. (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 18229	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
5		dmdprdsplitlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
6	4, 5	fssresd 5984	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
7		fdm 5964	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐴) = 𝐴)
8		dmdprdsplitlem.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
9		eqid 2610	. . . . . . 7 ⊢ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 18226	. . . . . 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 221	. . . 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 478	. . 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 792	. . . . . 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 474	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
17	16	ad2antrr 758	. . . . . . . . . 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 758	. . . . . . . . . 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 790	. . . . . . . . . 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 2610	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
22	9, 17, 19, 20, 21	dprdff 18234	. . . . . . . . 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 6159	. . . . . . . 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 758	. . . . . . . . . 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 5371	. . . . . . . . 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 4042	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝐴 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = (𝑓‘𝑛))
27	26	mpteq2ia 4668	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛))
28	25, 27	syl6eq 2660	. . . . . . . 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 2647	. . . . . . 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 6565	. . . . . 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 2610	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
32	2	ad2antrr 758	. . . . . . . 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 18227	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
34		grpmnd 17252	. . . . . . . 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 18223	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
37	36	ad2antrr 758	. . . . . . 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 758	. . . . . . . 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 796	. . . . . . . . . . . 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 18235	. . . . . . . . . . 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 6117	. . . . . . . . . . . 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 2690	. . . . . . . . . 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 758	. . . . . . . . . . . . 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 6267	. . . . . . . . . . . 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 17425	. . . . . . . . . . . 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 4070	. . . . . . . . 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 6389	. . . . . . . . . . . 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 758	. . . . . . . . . 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 5840	. . . . . . . . . . 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 18236	. . . . . . . . . 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 476	. . . . . . . . . . . . . . 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 3695	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 )) → ¬ 𝑛 ∈ (𝑓 supp 0 ))
61	60	ad2antlr 759	. . . . . . . . . . . . . . 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 3551	. . . . . . . . . . . . . 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 3587	. . . . . . . . . . . . . . . . 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 4733	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ V)
66	65	ad2antrr 758	. . . . . . . . . . . . . . . 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 6113	. . . . . . . . . . . . . . . . . 18 ⊢ (0g‘𝐺) ∈ V
68	8, 67	eqeltri 2684	. . . . . . . . . . . . . . . . 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 7213	. . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 4066	. . . . . . . . . . . 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 4075	. . . . . . . . . . . 12 ⊢ if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
75	73, 74	syl6eq 2660	. . . . . . . . . . 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 7216	. . . . . . . . . 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 8174	. . . . . . . . . 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 1319	. . . . . . . . 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 18233	. . . . . . . 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 18234	. . . . . . 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 18237	. . . . . . 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 3695	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
83	82	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)
84	83	iffalsed 4047	. . . . . . . 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 7216	. . . . . . 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 18133	. . . . . 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 2648	. . . . 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 758	. . . . . 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 18245	. . . . 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 221	. . . 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 6105	. . 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 3694	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → 𝑋 ∈ 𝐼)
94	93	ad2antlr 759	. . . 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 2676	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
96		fveq2 6103	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑓‘𝑛) = (𝑓‘𝑋))
97	95, 96	ifbieq1d 4059	. . . . 5 ⊢ (𝑛 = 𝑋 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
98		eqid 2610	. . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
99		fvex 6113	. . . . . 6 ⊢ (𝑓‘𝑛) ∈ V
100	99, 68	ifex 4106	. . . . 5 ⊢ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) ∈ V
101	97, 98, 100	fvmpt3i 6196	. . . 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 3695	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑋 ∈ 𝐴)
104	103	ad2antlr 759	. . . 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 4047	. . 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 2648	. 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 3017	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   ↾ cres 5040  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Xcixp 7794   finSupp cfsupp 8158  Basecbs 15695  0_gc0g 15923   Σ_g cgsu 15924  Mndcmnd 17117  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  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-cmn 18018  df-dprd 18217

This theorem is referenced by:  dprddisj2  18261