Theorem dprdfeq0 19137

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 2798	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 19127	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	feqmptd 6708	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
8	7	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
9	1, 2, 3, 4	dprdfcl 19128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
10	9	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
11		eldprdi.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐺)
12	2	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆)
13	3	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼)
14		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
15		eqid 2798	. . . . . . . . . . . . . 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 19132	. . . . . . . . . . . . 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 498	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊)
18	4	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝑊)
19		eqid 2798	. . . . . . . . . . . 12 ⊢ (-g‘𝐺) = (-g‘𝐺)
20	11, 1, 12, 13, 17, 18, 19	dprdfsub 19136	. . . . . . . . . . 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 499	. . . . . . . . . 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 19116	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ V)
23	22	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V)
24		fvex 6658	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑥) ∈ V
25	11	fvexi 6659	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
26	24, 25	ifex 4473	. . . . . . . . . . . . 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 6660	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ V)
29		eqidd 2799	. . . . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
31	30	feqmptd 6708	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 = (𝑦 ∈ 𝐼 ↦ (𝐹‘𝑦)))
32	23, 27, 28, 29, 31	offval2 7406	. . . . . . . . . . 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 7151	. . . . . . . . . 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 499	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥))
35		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝐹) = 0 )
36	34, 35	oveq12d 7153	. . . . . . . . . . 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 19120	. . . . . . . . . . . . 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 6829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺))
40	5, 11, 19	grpsubid1 18176	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
41	38, 39, 40	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
42	36, 41	eqtrd 2833	. . . . . . . . . 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 2841	. . . . . . . . 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 2798	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
45		grpmnd 18102	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
46	2, 37, 45	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
47	46	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd)
48	5	subgacs 18305	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49		acsmre 16915	. . . . . . . . . . . . 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 5907	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
52	2, 3	dprdf2 19122	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
53	52	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
54	53	frnd 6494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55		mresspw 16855	. . . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
58	51, 57	sstrid 3926	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59		sspwuni 4985	. . . . . . . . . . . . 13 ⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
60	58, 59	sylib 221	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61		eqid 2798	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
62	61	mrccl 16874	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63	50, 60, 62	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64		subgsubm 18293	. . . . . . . . . . 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 7142	. . . . . . . . . . . . 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 2874	. . . . . . . . . . . 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 7142	. . . . . . . . . . . . 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 2874	. . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
71	70	fveq2d 6649	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
72	71	oveq2d 7151	. . . . . . . . . . . . 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 18175	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
74	38, 39, 73	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
75	11	subg0cl 18279	. . . . . . . . . . . . . . . 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 2890	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
78	77	ad2antrr 725	. . . . . . . . . . . . 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 2890	. . . . . . . . . . . 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 725	. . . . . . . . . . . . 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 16888	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
83	82	ad2antrr 725	. . . . . . . . . . . . . 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 19128	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
85	84	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
86	53	ffnd 6488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆 Fn 𝐼)
87	86	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88		difssd 4060	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89		df-ne 2988	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥)
90		eldifsn 4680	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥))
91	90	biimpri 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
92	89, 91	sylan2br 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
93	92	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94		fnfvima 6973	. . . . . . . . . . . . . . . 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 4805	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑦) ∈ (𝑆‘𝑦) ∧ (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
97	85, 95, 96	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
98	83, 97	sseldd 3916	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
99	19	subgsubcl 18282	. . . . . . . . . . . . 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 4462	. . . . . . . . . . 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 6856	. . . . . . . . . 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 498	. . . . . . . . . . . 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 2891	. . . . . . . . . . 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 19130	. . . . . . . . . 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 19129	. . . . . . . . . 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 19031	. . . . . . . . 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 2891	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
109	10, 108	elind 4121	. . . . . . 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 19124	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111	109, 110	eleqtrd 2892	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ { 0 })
112		elsni 4542	. . . . . 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 5125	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐼 ↦ 0 ))
115	8, 114	eqtrd 2833	. . 3 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))
116	115	ex 416	. 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))
117	11	gsumz 17992	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
118	46, 22, 117	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
119		oveq2 7143	. . . 4 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )))
120	119	eqeq1d 2800	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 ))
121	118, 120	syl5ibrcom 250	. 2 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg 𝐹) = 0 ))
122	116, 121	impbid 215	1 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ifcif 4425  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ran crn 5520   “ cima 5522   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387  Xcixp 8444   finSupp cfsupp 8817  Basecbs 16475  0_gc0g 16705   Σ_g cgsu 16706  Moorecmre 16845  mrClscmrc 16846  ACScacs 16848  Mndcmnd 17903  SubMndcsubmnd 17947  Grpcgrp 18095  -_gcsg 18097  SubGrpcsubg 18265  Cntzccntz 18437   DProd cdprd 19108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-gim 18391  df-cntz 18439  df-oppg 18466  df-cmn 18900  df-dprd 19110

This theorem is referenced by:  dprdf11  19138