Theorem dprdfeq0 20289

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 2763	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
6	1, 2, 3, 4, 5	dprdff 20279	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺))
7	6	feqmptd 6984	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
8	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)))
9	1, 2, 3, 4	dprdfcl 20280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
10	9	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥))
11		eldprdi.0	. . . . . . . . . . . 12 ⊢ 0 = (0g‘𝐺)
12	2	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆)
13	3	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼)
14		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
15		eqid 2763	. . . . . . . . . . . . . 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 20284	. . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊)
18	4	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝑊)
19		eqid 2763	. . . . . . . . . . . 12 ⊢ (-g‘𝐺) = (-g‘𝐺)
20	11, 1, 12, 13, 17, 18, 19	dprdfsub 20288	. . . . . . . . . . 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 501	. . . . . . . . . 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 20268	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ V)
23	22	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V)
24		fvex 6929	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑥) ∈ V
25	11	fvexi 6930	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
26	24, 25	ifex 4536	. . . . . . . . . . . . 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 6931	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ V)
29		eqidd 2764	. . . . . . . . . . . 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 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
31	30	feqmptd 6984	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 = (𝑦 ∈ 𝐼 ↦ (𝐹‘𝑦)))
32	23, 27, 28, 29, 31	offval2 7745	. . . . . . . . . . 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 7467	. . . . . . . . . 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 501	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥))
35		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝐹) = 0 )
36	34, 35	oveq12d 7469	. . . . . . . . . . 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 20272	. . . . . . . . . . . . 13 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp)
38	12, 37	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp)
39	30, 14	ffvelcdmd 7117	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺))
40	5, 11, 19	grpsubid1 19285	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
41	38, 39, 40	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥))
42	36, 41	eqtrd 2798	. . . . . . . . . 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 2806	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
45		grpmnd 19201	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
46	2, 37, 45	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
47	46	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd)
48	5	subgacs 19421	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49		acsmre 17876	. . . . . . . . . . . . 13 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
50	38, 48, 49	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
51		imassrn 6216	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
52	2, 3	dprdf2 20274	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
53	52	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
54	53	frnd 6748	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55		mresspw 17812	. . . . . . . . . . . . . . . 16 ⊢ ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
56	50, 55	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
57	54, 56	sstrd 3923	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
58	51, 57	sstrid 3924	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59		sspwuni 5065	. . . . . . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
62	61	mrccl 17835	. . . . . . . . . . . 12 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
63	50, 60, 62	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64		subgsubm 19409	. . . . . . . . . . 11 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
65	63, 64	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
66		oveq1 7458	. . . . . . . . . . . . 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 2849	. . . . . . . . . . . 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 7458	. . . . . . . . . . . . 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 2849	. . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
71	70	fveq2d 6920	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥))
72	71	oveq2d 7467	. . . . . . . . . . . . 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 19284	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
74	38, 39, 73	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 )
75	11	subg0cl 19394	. . . . . . . . . . . . . . . 16 ⊢ (((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
76	63, 75	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
77	74, 76	eqeltrd 2864	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
78	77	ad2antrr 739	. . . . . . . . . . . . 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 2864	. . . . . . . . . . . 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 739	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
81	80, 75	syl 18	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
82	50, 61, 60	mrcssidd 17849	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
83	82	ad2antrr 739	. . . . . . . . . . . . . 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 20280	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
85	84	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ (𝑆‘𝑦))
86	53	ffnd 6740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆 Fn 𝐼)
87	86	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88		difssd 4076	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89		df-ne 2960	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥)
90		eldifsn 4750	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥))
91	90	biimpri 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
92	89, 91	sylan2br 607	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
93	92	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94		fnfvima 7270	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼 ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
95	87, 88, 93, 94	syl3anc 1399	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96		elunii 4877	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑦) ∈ (𝑆‘𝑦) ∧ (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
97	85, 95, 96	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥})))
98	83, 97	sseldd 3914	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
99	19	subgsubcl 19398	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 4524	. . . . . . . . . . 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 7147	. . . . . . . . . 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 500	. . . . . . . . . . . 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 2865	. . . . . . . . . . 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 20282	. . . . . . . . . 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 20281	. . . . . . . . . 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 20183	. . . . . . . . 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 2865	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))))
109	10, 108	elind 4100	. . . . . . 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 20276	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111	109, 110	eleqtrd 2866	. . . . . 6 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ { 0 })
112		elsni 4601	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ { 0 } → (𝐹‘𝑥) = 0 )
113	111, 112	syl 18	. . . . 5 ⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = 0 )
114	113	mpteq2dva 5204	. . . 4 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐼 ↦ 0 ))
115	8, 114	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))
116	115	ex 418	. 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )))
117	11	gsumz 19082	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
118	46, 22, 117	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )
119		oveq2 7459	. . . 4 ⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )))
120	119	eqeq1d 2765	. . 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 )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ifcif 4485  𝒫 cpw 4560  {csn 4588  ∪ cuni 4872   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   “ cima 5661   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∘_f cof 7723  Xcixp 8963   finSupp cfsupp 9391  Basecbs 17431  0_gc0g 17660   Σ_g cgsu 17661  Moorecmre 17802  mrClscmrc 17803  ACScacs 17805  Mndcmnd 18973  SubMndcsubmnd 19027  Grpcgrp 19194  -_gcsg 19196  SubGrpcsubg 19380  Cntzccntz 19580   DProd cdprd 20260

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-ixp 8964  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-oi 9541  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-fzo 13829  df-seq 14185  df-hash 14516  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-0g 17662  df-gsum 17663  df-mre 17806  df-mrc 17807  df-acs 17809  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-mhm 19028  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-mulg 19328  df-subg 19383  df-ghm 19478  df-gim 19523  df-cntz 19582  df-oppg 19611  df-cmn 20047  df-dprd 20262

This theorem is used by:  dprdf11  20290