Theorem dprdfeq0 20022

Description: The zero function is the only function that sums to zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

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

Assertion

Ref	Expression
dprdfeq0	⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))

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

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

Proof of Theorem dprdfeq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldprdi.w	. . . . . . 7 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
2		eldprdi.1	. . . . . . 7 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
3		eldprdi.2	. . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼)
4		eldprdi.3	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
5		eqid 2726	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 20012	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	feqmptd 6971	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
8	7	adantr 479	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
9	1, 2, 3, 4	dprdfcl 20013	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
10	9	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
11		eldprdi.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐺)
12	2	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆)
13	3	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼)
14		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
15		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))
16	11, 1, 12, 13, 14, 10, 15	dprdfid 20017	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥)))
17	16	simpld 493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊)
18	4	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝑊)
19		eqid 2726	. . . . . . . . . . . 12 ⊢ (-g‘𝐺) = (-g‘𝐺)
20	11, 1, 12, 13, 17, 18, 19	dprdfsub 20021	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f (-g‘𝐺)𝐹) ∈ 𝑊 ∧ (𝐺 Σg ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f (-g‘𝐺)𝐹)) = ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-g‘𝐺)(𝐺 Σg 𝐹))))
21	20	simprd 494	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f (-g‘𝐺)𝐹)) = ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-g‘𝐺)(𝐺 Σg 𝐹)))
22	2, 3	dprddomcld 20001	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ V)
23	22	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V)
24		fvex 6914	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑥) ∈ V
25	11	fvexi 6915	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
26	24, 25	ifex 4583	. . . . . . . . . . . . 13 ⊢ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) ∈ V
27	26	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) ∈ V)
28		fvexd 6916	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ V)
29		eqidd 2727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))
30	6	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
31	30	feqmptd 6971	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 = (𝑦 ∈ 𝐼 ↦ (𝐹‘𝑦)))
32	23, 27, 28, 29, 31	offval2 7710	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f (-g‘𝐺)𝐹) = (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦))))
33	32	oveq2d 7440	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f (-g‘𝐺)𝐹)) = (𝐺 Σg (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)))))
34	16	simprd 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥))
35		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝐹) = 0 )
36	34, 35	oveq12d 7442	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-g‘𝐺)(𝐺 Σg 𝐹)) = ((𝐹‘𝑥)(-g‘𝐺) 0 ))
37		dprdgrp 20005	. . . . . . . . . . . . 13 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
38	12, 37	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
39	30, 14	ffvelcdmd 7099	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺))
40	5, 11, 19	grpsubid1 19019	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
41	38, 39, 40	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
42	36, 41	eqtrd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-g‘𝐺)(𝐺 Σg 𝐹)) = (𝐹‘𝑥))
43	21, 33, 42	3eqtr3d 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)))) = (𝐹‘𝑥))
44		eqid 2726	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
45		grpmnd 18935	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
46	2, 37, 45	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
47	46	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd)
48	5	subgacs 19155	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49		acsmre 17665	. . . . . . . . . . . . 13 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
50	38, 48, 49	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
51		imassrn 6080	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
52	2, 3	dprdf2 20007	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
53	52	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
54	53	frnd 6736	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55		mresspw 17605	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
56	50, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
57	54, 56	sstrd 3990	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
58	51, 57	sstrid 3991	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59		sspwuni 5108	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
60	58, 59	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61		eqid 2726	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
62	61	mrccl 17624	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63	50, 60, 62	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64		subgsubm 19142	. . . . . . . . . . 11 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
65	63, 64	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
66		oveq1 7431	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) = (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)))
67	66	eleq1d 2811	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → (((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ↔ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
68		oveq1 7431	. . . . . . . . . . . . 13 ⊢ ( 0 = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → ( 0 (-g‘𝐺)(𝐹‘𝑦)) = (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)))
69	68	eleq1d 2811	. . . . . . . . . . . 12 ⊢ ( 0 = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → (( 0 (-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ↔ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
70		simpr 483	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
71	70	fveq2d 6905	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
72	71	oveq2d 7440	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)))
73	5, 11, 19	grpsubid 19018	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
74	38, 39, 73	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
75	11	subg0cl 19128	. . . . . . . . . . . . . . . 16 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
76	63, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
77	74, 76	eqeltrd 2826	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
78	77	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
79	72, 78	eqeltrd 2826	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
80	63	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
81	80, 75	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
82	50, 61, 60	mrcssidd 17638	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
83	82	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
84	1, 12, 13, 18	dprdfcl 20013	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
85	84	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
86	53	ffnd 6729	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆 Fn 𝐼)
87	86	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88		difssd 4132	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89		df-ne 2931	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥)
90		eldifsn 4795	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥))
91	90	biimpri 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
92	89, 91	sylan2br 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
93	92	adantll 712	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94		fnfvima 7250	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼 ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
95	87, 88, 93, 94	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96		elunii 4918	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑦) ∈ (𝑆‘𝑦) ∧ (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
97	85, 95, 96	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
98	83, 97	sseldd 3980	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
99	19	subgsubcl 19131	. . . . . . . . . . . . 13 ⊢ ((((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∧ (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) → ( 0 (-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
100	80, 81, 98, 99	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ( 0 (-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
101	67, 69, 79, 100	ifbothda 4571	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
102	101	fmpttd 7129	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
103	20	simpld 493	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f (-g‘𝐺)𝐹) ∈ 𝑊)
104	32, 103	eqeltrrd 2827	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦))) ∈ 𝑊)
105	1, 12, 13, 104, 44	dprdfcntz 20015	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦))) ⊆ ((Cntz‘𝐺)‘ran (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)))))
106	1, 12, 13, 104	dprdffsupp 20014	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦))) finSupp 0 )
107	11, 44, 47, 23, 65, 102, 105, 106	gsumzsubmcl 19916	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-g‘𝐺)(𝐹‘𝑦)))) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
108	43, 107	eqeltrrd 2827	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
109	10, 108	elind 4195	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
110	12, 13, 14, 11, 61	dprddisj 20009	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111	109, 110	eleqtrd 2828	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ { 0 })
112		elsni 4650	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ { 0 } → (𝐹‘𝑥) = 0 )
113	111, 112	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = 0 )
114	113	mpteq2dva 5253	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐼 ↦ 0 ))
115	8, 114	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))
116	115	ex 411	. 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))
117	11	gsumz 18826	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
118	46, 22, 117	syl2anc 582	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
119		oveq2 7432	. . . 4 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )))
120	119	eqeq1d 2728	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 ))
121	118, 120	syl5ibrcom 246	. 2 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg 𝐹) = 0 ))
122	116, 121	impbid 211	1 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  {crab 3419  Vcvv 3462   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ifcif 4533  𝒫 cpw 4607  {csn 4633  ∪ cuni 4913   class class class wbr 5153   ↦ cmpt 5236  dom cdm 5682  ran crn 5683   “ cima 5685   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424   ∘_f cof 7688  Xcixp 8926   finSupp cfsupp 9405  Basecbs 17213  0_gc0g 17454   Σ_g cgsu 17455  Moorecmre 17595  mrClscmrc 17596  ACScacs 17598  Mndcmnd 18727  SubMndcsubmnd 18772  Grpcgrp 18928  -_gcsg 18930  SubGrpcsubg 19114  Cntzccntz 19309   DProd cdprd 19993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-0g 17456  df-gsum 17457  df-mre 17599  df-mrc 17600  df-acs 17602  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-subg 19117  df-ghm 19207  df-gim 19253  df-cntz 19311  df-oppg 19340  df-cmn 19780  df-dprd 19995

This theorem is referenced by:  dprdf11  20023