Theorem dmdprdsplitlem 18903

Description: Lemma for dmdprdsplit 18913. (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 18873	. . . . . . 7 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
5		dmdprdsplitlem.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
6	4, 5	fssresd 6368	. . . . . 6 ⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺))
7		fdm 6346	. . . . . 6 ⊢ ((𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐴) = 𝐴)
8		dmdprdsplitlem.0	. . . . . . 7 ⊢ 0 = (0g‘𝐺)
9		eqid 2772	. . . . . . 7 ⊢ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 18870	. . . . . 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 224	. . . 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 488	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
14	13	adantr 473	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐴 ((𝑆 ↾ 𝐴)‘𝑖) ∣ ℎ finSupp 0 } (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
15		simprr 760	. . . . . 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 487	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴))
17	16	ad2antrr 713	. . . . . . . . . 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 713	. . . . . . . . . 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 758	. . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
22	9, 17, 19, 20, 21	dprdff 18878	. . . . . . . . 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 6556	. . . . . . . 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 713	. . . . . . . . . 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 5747	. . . . . . . . 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 4350	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝐴 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = (𝑓‘𝑛))
27	26	mpteq2ia 5012	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐴 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐴 ↦ (𝑓‘𝑛))
28	25, 27	syl6eq 2824	. . . . . . . 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 2811	. . . . . . 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 6986	. . . . . 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 2772	. . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
32	2	ad2antrr 713	. . . . . . . 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 18871	. . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
34		grpmnd 17892	. . . . . . . 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 18867	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ V)
37	36	ad2antrr 713	. . . . . . 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 713	. . . . . . . 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 473	. . . . . . . . . . . 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 473	. . . . . . . . . . . 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 764	. . . . . . . . . . . 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 18879	. . . . . . . . . . 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 6512	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑛) = (𝑆‘𝑛))
45	44	adantl 474	. . . . . . . . . . 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 2862	. . . . . . . . . 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 713	. . . . . . . . . . . . 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 6670	. . . . . . . . . . . 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 18065	. . . . . . . . . . . 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 473	. . . . . . . . . 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 4378	. . . . . . . . 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 6807	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) ∈ V)
54	53	ad2antrr 713	. . . . . . . . . 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 6220	. . . . . . . . . . 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 18880	. . . . . . . . . 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 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)
59		eldifn 3988	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (𝐼 ∖ (𝑓 supp 0 )) → ¬ 𝑛 ∈ (𝑓 supp 0 ))
60	59	ad2antlr 714	. . . . . . . . . . . . . . 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 3834	. . . . . . . . . . . . . 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 3874	. . . . . . . . . . . . . . . 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 5078	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ V)
64	63	ad2antrr 713	. . . . . . . . . . . . . . . 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 6507	. . . . . . . . . . . . . . . . 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 7658	. . . . . . . . . . . . . . 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 702	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 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 4374	. . . . . . . . . . . 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 4383	. . . . . . . . . . . 12 ⊢ if(𝑛 ∈ 𝐴, 0 , 0 ) = 0
72	70, 71	syl6eq 2824	. . . . . . . . . . 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 7661	. . . . . . . . . 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 8638	. . . . . . . . . 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 826	. . . . . . . . 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 18877	. . . . . . . 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 18878	. . . . . . 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 18881	. . . . . . 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 3988	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑛 ∈ 𝐴)
80	79	adantl 474	. . . . . . . . 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 4355	. . . . . . . 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 7661	. . . . . . 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 18777	. . . . . 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 2812	. . . . 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 713	. . . . . 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 18889	. . . . 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 224	. . . 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 6495	. . 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 3987	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → 𝑋 ∈ 𝐼)
91	90	ad2antlr 714	. . . 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 2847	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑛 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
93		fveq2 6493	. . . . . 6 ⊢ (𝑛 = 𝑋 → (𝑓‘𝑛) = (𝑓‘𝑋))
94	92, 93	ifbieq1d 4367	. . . . 5 ⊢ (𝑛 = 𝑋 → if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) = if(𝑋 ∈ 𝐴, (𝑓‘𝑋), 0 ))
95		eqid 2772	. . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ))
96		fvex 6506	. . . . . 6 ⊢ (𝑓‘𝑛) ∈ V
97	96, 65	ifex 4392	. . . . 5 ⊢ if(𝑛 ∈ 𝐴, (𝑓‘𝑛), 0 ) ∈ V
98	94, 95, 97	fvmpt3i 6594	. . . 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 3988	. . . . 5 ⊢ (𝑋 ∈ (𝐼 ∖ 𝐴) → ¬ 𝑋 ∈ 𝐴)
101	100	ad2antlr 714	. . . 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 4355	. . 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 2812	. 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 3230	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐼 ∖ 𝐴)) → (𝐹‘𝑋) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∃wrex 3083  {crab 3086  Vcvv 3409   ∖ cdif 3820   ⊆ wss 3823  ifcif 4344   class class class wbr 4923   ↦ cmpt 5002  dom cdm 5401   ↾ cres 5403  Fun wfun 6176  ⟶wf 6178  ‘cfv 6182  (class class class)co 6970   supp csupp 7627  Xcixp 8253   finSupp cfsupp 8622  Basecbs 16333  0_gc0g 16563   Σ_g cgsu 16564  Mndcmnd 17756  Grpcgrp 17885  SubGrpcsubg 18051  Cntzccntz 18210   DProd cdprd 18859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-se 5361  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7495  df-2nd 7496  df-supp 7628  df-tpos 7689  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-ixp 8254  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-fsupp 8623  df-oi 8763  df-card 9156  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-n0 11702  df-z 11788  df-uz 12053  df-fz 12703  df-fzo 12844  df-seq 13179  df-hash 13500  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-ress 16341  df-plusg 16428  df-0g 16565  df-gsum 16566  df-mre 16709  df-mrc 16710  df-acs 16712  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-mhm 17797  df-submnd 17798  df-grp 17888  df-minusg 17889  df-sbg 17890  df-mulg 18006  df-subg 18054  df-ghm 18121  df-gim 18164  df-cntz 18212  df-oppg 18239  df-cmn 18662  df-dprd 18861

This theorem is referenced by:  dprddisj2  18905