Theorem dprdfeq0 19625

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 2738	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 19615	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	feqmptd 6837	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
8	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
9	1, 2, 3, 4	dprdfcl 19616	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
10	9	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
11		eldprdi.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐺)
12	2	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆)
13	3	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼)
14		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
15		eqid 2738	. . . . . . . . . . . . . 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 19620	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊)
18	4	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝑊)
19		eqid 2738	. . . . . . . . . . . 12 ⊢ (-g‘𝐺) = (-g‘𝐺)
20	11, 1, 12, 13, 17, 18, 19	dprdfsub 19624	. . . . . . . . . . 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 496	. . . . . . . . . 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 19604	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ V)
23	22	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V)
24		fvex 6787	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑥) ∈ V
25	11	fvexi 6788	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
26	24, 25	ifex 4509	. . . . . . . . . . . . 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 6789	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ V)
29		eqidd 2739	. . . . . . . . . . . 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 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
31	30	feqmptd 6837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 = (𝑦 ∈ 𝐼 ↦ (𝐹‘𝑦)))
32	23, 27, 28, 29, 31	offval2 7553	. . . . . . . . . . 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 7291	. . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥))
35		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝐹) = 0 )
36	34, 35	oveq12d 7293	. . . . . . . . . . 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 19608	. . . . . . . . . . . . 13 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
38	12, 37	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
39	30, 14	ffvelrnd 6962	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺))
40	5, 11, 19	grpsubid1 18660	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
41	38, 39, 40	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
42	36, 41	eqtrd 2778	. . . . . . . . . 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 2786	. . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
45		grpmnd 18584	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
46	2, 37, 45	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
47	46	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd)
48	5	subgacs 18789	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49		acsmre 17361	. . . . . . . . . . . . 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 5980	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
52	2, 3	dprdf2 19610	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
53	52	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
54	53	frnd 6608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55		mresspw 17301	. . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
58	51, 57	sstrid 3932	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59		sspwuni 5029	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
62	61	mrccl 17320	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63	50, 60, 62	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64		subgsubm 18777	. . . . . . . . . . 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 7282	. . . . . . . . . . . . 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 2823	. . . . . . . . . . . 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 7282	. . . . . . . . . . . . 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 2823	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
71	70	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
72	71	oveq2d 7291	. . . . . . . . . . . . 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 18659	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
74	38, 39, 73	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
75	11	subg0cl 18763	. . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
78	77	ad2antrr 723	. . . . . . . . . . . . 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 2839	. . . . . . . . . . . 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 723	. . . . . . . . . . . . 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 17334	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
83	82	ad2antrr 723	. . . . . . . . . . . . . 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 19616	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
85	84	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
86	53	ffnd 6601	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆 Fn 𝐼)
87	86	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88		difssd 4067	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89		df-ne 2944	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥)
90		eldifsn 4720	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥))
91	90	biimpri 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
92	89, 91	sylan2br 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
93	92	adantll 711	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94		fnfvima 7109	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼 ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
95	87, 88, 93, 94	syl3anc 1370	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96		elunii 4844	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑦) ∈ (𝑆‘𝑦) ∧ (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
97	85, 95, 96	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
98	83, 97	sseldd 3922	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
99	19	subgsubcl 18766	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 4497	. . . . . . . . . . 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 6989	. . . . . . . . . 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 495	. . . . . . . . . . . 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 2840	. . . . . . . . . . 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 19618	. . . . . . . . . 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 19617	. . . . . . . . . 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 19519	. . . . . . . . 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 2840	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
109	10, 108	elind 4128	. . . . . . 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 19612	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111	109, 110	eleqtrd 2841	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ { 0 })
112		elsni 4578	. . . . . 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 5174	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐼 ↦ 0 ))
115	8, 114	eqtrd 2778	. . 3 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))
116	115	ex 413	. 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))
117	11	gsumz 18474	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
118	46, 22, 117	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
119		oveq2 7283	. . . 4 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )))
120	119	eqeq1d 2740	. . 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 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ifcif 4459  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531  Xcixp 8685   finSupp cfsupp 9128  Basecbs 16912  0_gc0g 17150   Σ_g cgsu 17151  Moorecmre 17291  mrClscmrc 17292  ACScacs 17294  Mndcmnd 18385  SubMndcsubmnd 18429  Grpcgrp 18577  -_gcsg 18579  SubGrpcsubg 18749  Cntzccntz 18921   DProd cdprd 19596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-mulg 18701  df-subg 18752  df-ghm 18832  df-gim 18875  df-cntz 18923  df-oppg 18950  df-cmn 19388  df-dprd 19598

This theorem is referenced by:  dprdf11  19626