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Theorem dprdfeq0 19934

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 = (0_g‘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 Σ_g 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 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 19924	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	feqmptd 6961	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
9	1, 2, 3, 4	dprdfcl 19925	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
10	9	adantlr 712	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
11		eldprdi.0	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐺)
12	2	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆)
13	3	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼)
14		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
15		eqid 2731	. . . . . . . . . . . . . 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 19929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥)))
17	16	simpld 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊)
18	4	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝑊)
19		eqid 2731	. . . . . . . . . . . 12 ⊢ (-_g‘𝐺) = (-_g‘𝐺)
20	11, 1, 12, 13, 17, 18, 19	dprdfsub 19933	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘_f (-_g‘𝐺)𝐹) ∈ 𝑊 ∧ (𝐺 Σ_g ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘_f (-_g‘𝐺)𝐹)) = ((𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-_g‘𝐺)(𝐺 Σ_g 𝐹))))
21	20	simprd 495	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σ_g ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘_f (-_g‘𝐺)𝐹)) = ((𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-_g‘𝐺)(𝐺 Σ_g 𝐹)))
22	2, 3	dprddomcld 19913	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ V)
23	22	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V)
24		fvex 6905	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑥) ∈ V
25	11	fvexi 6906	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
26	24, 25	ifex 4579	. . . . . . . . . . . . 13 ⊢ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) ∈ V
27	26	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) ∈ V)
28		fvexd 6907	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ V)
29		eqidd 2732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 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 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
31	30	feqmptd 6961	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 = (𝑦 ∈ 𝐼 ↦ (𝐹‘𝑦)))
32	23, 27, 28, 29, 31	offval2 7693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σ_g 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 7428	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σ_g ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘_f (-_g‘𝐺)𝐹)) = (𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-_g‘𝐺)(𝐹‘𝑦)))))
34	16	simprd 495	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥))
35		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σ_g 𝐹) = 0 )
36	34, 35	oveq12d 7430	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-_g‘𝐺)(𝐺 Σ_g 𝐹)) = ((𝐹‘𝑥)(-_g‘𝐺) 0 ))
37		dprdgrp 19917	. . . . . . . . . . . . 13 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
38	12, 37	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
39	30, 14	ffvelcdmd 7088	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺))
40	5, 11, 19	grpsubid1 18945	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-_g‘𝐺) 0 ) = (𝐹‘𝑥))
41	38, 39, 40	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-_g‘𝐺) 0 ) = (𝐹‘𝑥))
42	36, 41	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )))(-_g‘𝐺)(𝐺 Σ_g 𝐹)) = (𝐹‘𝑥))
43	21, 33, 42	3eqtr3d 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-_g‘𝐺)(𝐹‘𝑦)))) = (𝐹‘𝑥))
44		eqid 2731	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
45		grpmnd 18863	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
46	2, 37, 45	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
47	46	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd)
48	5	subgacs 19078	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49		acsmre 17601	. . . . . . . . . . . . 13 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
50	38, 48, 49	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
51		imassrn 6071	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
52	2, 3	dprdf2 19919	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
53	52	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
54	53	frnd 6726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55		mresspw 17541	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
56	50, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
57	54, 56	sstrd 3993	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
58	51, 57	sstrid 3994	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59		sspwuni 5104	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
60	58, 59	sylib 217	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61		eqid 2731	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
62	61	mrccl 17560	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63	50, 60, 62	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64		subgsubm 19065	. . . . . . . . . . 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 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
66		oveq1 7419	. . . . . . . . . . . . 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 2817	. . . . . . . . . . . 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 7419	. . . . . . . . . . . . 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 2817	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
71	70	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
72	71	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 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 18944	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-_g‘𝐺)(𝐹‘𝑥)) = 0 )
74	38, 39, 73	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-_g‘𝐺)(𝐹‘𝑥)) = 0 )
75	11	subg0cl 19051	. . . . . . . . . . . . . . . 16 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
76	63, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
77	74, 76	eqeltrd 2832	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-_g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
78	77	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-_g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
79	72, 78	eqeltrd 2832	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 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 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
81	80, 75	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
82	50, 61, 60	mrcssidd 17574	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
83	82	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
84	1, 12, 13, 18	dprdfcl 19925	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
85	84	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
86	53	ffnd 6719	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆 Fn 𝐼)
87	86	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88		difssd 4133	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89		df-ne 2940	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥)
90		eldifsn 4791	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥))
91	90	biimpri 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
92	89, 91	sylan2br 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
93	92	adantll 711	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94		fnfvima 7238	. . . . . . . . . . . . . . . 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 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96		elunii 4914	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑦) ∈ (𝑆‘𝑦) ∧ (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
97	85, 95, 96	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
98	83, 97	sseldd 3984	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
99	19	subgsubcl 19054	. . . . . . . . . . . . 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 Σ_g 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 4567	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐺 Σ_g 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 7117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 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 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘_f (-_g‘𝐺)𝐹) ∈ 𝑊)
104	32, 103	eqeltrrd 2833	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-_g‘𝐺)(𝐹‘𝑦))) ∈ 𝑊)
105	1, 12, 13, 104, 44	dprdfcntz 19927	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 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 19926	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σ_g 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 19828	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σ_g (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )(-_g‘𝐺)(𝐹‘𝑦)))) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
108	43, 107	eqeltrrd 2833	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
109	10, 108	elind 4195	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))))
110	12, 13, 14, 11, 61	dprddisj 19921	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111	109, 110	eleqtrd 2834	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ { 0 })
112		elsni 4646	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ { 0 } → (𝐹‘𝑥) = 0 )
113	111, 112	syl 17	. . . . 5 ⊢ (((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = 0 )
114	113	mpteq2dva 5249	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐼 ↦ 0 ))
115	8, 114	eqtrd 2771	. . 3 ⊢ ((𝜑 ∧ (𝐺 Σ_g 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))
116	115	ex 412	. 2 ⊢ (𝜑 → ((𝐺 Σ_g 𝐹) = 0 → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))
117	11	gsumz 18754	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
118	46, 22, 117	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
119		oveq2 7420	. . . 4 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σ_g 𝐹) = (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 0 )))
120	119	eqeq1d 2733	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → ((𝐺 Σ_g 𝐹) = 0 ↔ (𝐺 Σ_g (𝑥 ∈ 𝐼 ↦ 0 )) = 0 ))
121	118, 120	syl5ibrcom 246	. 2 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σ_g 𝐹) = 0 ))
122	116, 121	impbid 211	1 ⊢ (𝜑 → ((𝐺 Σ_g 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 {crab 3431 Vcvv 3473 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ifcif 4529 𝒫 cpw 4603 {csn 4629 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ran crn 5678 “ cima 5680 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 ∘_f cof 7671 Xcixp 8894 finSupp cfsupp 9364 Basecbs 17149 0_gc0g 17390 Σ_g cgsu 17391 Moorecmre 17531 mrClscmrc 17532 ACScacs 17534 Mndcmnd 18660 SubMndcsubmnd 18705 Grpcgrp 18856 -_gcsg 18858 SubGrpcsubg 19037 Cntzccntz 19221 DProd cdprd 19905

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-0g 17392 df-gsum 17393 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-ghm 19129 df-gim 19174 df-cntz 19223 df-oppg 19252 df-cmn 19692 df-dprd 19907

This theorem is referenced by: dprdf11 19935

W3C validator