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Theorem dprdfeq0 19217
 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 = (0g𝐺)
eldprdi.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
eldprdi.1 (𝜑𝐺dom DProd 𝑆)
eldprdi.2 (𝜑 → dom 𝑆 = 𝐼)
eldprdi.3 (𝜑𝐹𝑊)
Assertion
Ref Expression
dprdfeq0 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
Distinct variable groups:   𝑥,,𝐹   ,𝑖,𝐺,𝑥   ,𝐼,𝑖,𝑥   𝜑,𝑥   0 ,,𝑥   𝑆,,𝑖,𝑥
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝑊(𝑥,,𝑖)   0 (𝑖)

Proof of Theorem dprdfeq0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldprdi.w . . . . . . 7 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
2 eldprdi.1 . . . . . . 7 (𝜑𝐺dom DProd 𝑆)
3 eldprdi.2 . . . . . . 7 (𝜑 → dom 𝑆 = 𝐼)
4 eldprdi.3 . . . . . . 7 (𝜑𝐹𝑊)
5 eqid 2758 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 19207 . . . . . 6 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76feqmptd 6725 . . . . 5 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
87adantr 484 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
91, 2, 3, 4dprdfcl 19208 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
109adantlr 714 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
11 eldprdi.0 . . . . . . . . . . . 12 0 = (0g𝐺)
122ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺dom DProd 𝑆)
133ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → dom 𝑆 = 𝐼)
14 simpr 488 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
15 eqid 2758 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))
1611, 1, 12, 13, 14, 10, 15dprdfid 19212 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥)))
1716simpld 498 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊)
184ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹𝑊)
19 eqid 2758 . . . . . . . . . . . 12 (-g𝐺) = (-g𝐺)
2011, 1, 12, 13, 17, 18, 19dprdfsub 19216 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹) ∈ 𝑊 ∧ (𝐺 Σg ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹)) = ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹))))
2120simprd 499 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹)) = ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)))
222, 3dprddomcld 19196 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ V)
2322ad2antrr 725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐼 ∈ V)
24 fvex 6675 . . . . . . . . . . . . . 14 (𝐹𝑥) ∈ V
2511fvexi 6676 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4473 . . . . . . . . . . . . 13 if(𝑦 = 𝑥, (𝐹𝑥), 0 ) ∈ V
2726a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → if(𝑦 = 𝑥, (𝐹𝑥), 0 ) ∈ V)
28 fvexd 6677 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ V)
29 eqidd 2759 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))
306ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
3130feqmptd 6725 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹 = (𝑦𝐼 ↦ (𝐹𝑦)))
3223, 27, 28, 29, 31offval2 7429 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹) = (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))))
3332oveq2d 7171 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹)) = (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))))
3416simprd 499 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥))
35 simplr 768 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg 𝐹) = 0 )
3634, 35oveq12d 7173 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = ((𝐹𝑥)(-g𝐺) 0 ))
37 dprdgrp 19200 . . . . . . . . . . . . 13 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
3812, 37syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Grp)
3930, 14ffvelrnd 6848 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (Base‘𝐺))
405, 11, 19grpsubid1 18256 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4138, 39, 40syl2anc 587 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4236, 41eqtrd 2793 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = (𝐹𝑥))
4321, 33, 423eqtr3d 2801 . . . . . . . . 9 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))) = (𝐹𝑥))
44 eqid 2758 . . . . . . . . . 10 (Cntz‘𝐺) = (Cntz‘𝐺)
45 grpmnd 18181 . . . . . . . . . . . 12 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
4746ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Mnd)
485subgacs 18385 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49 acsmre 16986 . . . . . . . . . . . . 13 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
5038, 48, 493syl 18 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
51 imassrn 5916 . . . . . . . . . . . . . 14 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
522, 3dprdf2 19202 . . . . . . . . . . . . . . . . 17 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
5352ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
5453frnd 6509 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55 mresspw 16926 . . . . . . . . . . . . . . . 16 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5650, 55syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5754, 56sstrd 3904 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5851, 57sstrid 3905 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59 sspwuni 4990 . . . . . . . . . . . . 13 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
6058, 59sylib 221 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61 eqid 2758 . . . . . . . . . . . . 13 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6261mrccl 16945 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
6350, 60, 62syl2anc 587 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64 subgsubm 18373 . . . . . . . . . . 11 (((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
6563, 64syl 17 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺))
66 oveq1 7162 . . . . . . . . . . . . 13 ((𝐹𝑥) = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6766eleq1d 2836 . . . . . . . . . . . 12 ((𝐹𝑥) = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → (((𝐹𝑥)(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ↔ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
68 oveq1 7162 . . . . . . . . . . . . 13 ( 0 = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ( 0 (-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6968eleq1d 2836 . . . . . . . . . . . 12 ( 0 = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → (( 0 (-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ↔ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
70 simpr 488 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
7170fveq2d 6666 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
7271oveq2d 7171 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = ((𝐹𝑥)(-g𝐺)(𝐹𝑥)))
735, 11, 19grpsubid 18255 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7438, 39, 73syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7511subg0cl 18359 . . . . . . . . . . . . . . . 16 (((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7663, 75syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7774, 76eqeltrd 2852 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7877ad2antrr 725 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7972, 78eqeltrd 2852 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8063ad2antrr 725 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
8180, 75syl 17 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8250, 61, 60mrcssidd 16959 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8382ad2antrr 725 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
841, 12, 13, 18dprdfcl 19208 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
8584adantr 484 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆𝑦))
8653ffnd 6503 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆 Fn 𝐼)
8786ad2antrr 725 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88 difssd 4040 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89 df-ne 2952 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
90 eldifsn 4680 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦𝐼𝑦𝑥))
9190biimpri 231 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐼𝑦𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9289, 91sylan2br 597 . . . . . . . . . . . . . . . . 17 ((𝑦𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9392adantll 713 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94 fnfvima 6992 . . . . . . . . . . . . . . . 16 ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9587, 88, 93, 94syl3anc 1368 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96 elunii 4806 . . . . . . . . . . . . . . 15 (((𝐹𝑦) ∈ (𝑆𝑦) ∧ (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9785, 95, 96syl2anc 587 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9883, 97sseldd 3895 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
9919subgsubcl 18362 . . . . . . . . . . . . 13 ((((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ 0 ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∧ (𝐹𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) → ( 0 (-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10080, 81, 98, 99syl3anc 1368 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → ( 0 (-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10167, 69, 79, 100ifbothda 4461 . . . . . . . . . . 11 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
102101fmpttd 6875 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10320simpld 498 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹) ∈ 𝑊)
10432, 103eqeltrrd 2853 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) ∈ 𝑊)
1051, 12, 13, 104, 44dprdfcntz 19210 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) ⊆ ((Cntz‘𝐺)‘ran (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))))
1061, 12, 13, 104dprdffsupp 19209 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) finSupp 0 )
10711, 44, 47, 23, 65, 102, 105, 106gsumzsubmcl 19111 . . . . . . . . 9 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10843, 107eqeltrrd 2853 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10910, 108elind 4101 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
11012, 13, 14, 11, 61dprddisj 19204 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111109, 110eleqtrd 2854 . . . . . 6 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ { 0 })
112 elsni 4542 . . . . . 6 ((𝐹𝑥) ∈ { 0 } → (𝐹𝑥) = 0 )
113111, 112syl 17 . . . . 5 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) = 0 )
114113mpteq2dva 5130 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥𝐼 ↦ (𝐹𝑥)) = (𝑥𝐼0 ))
1158, 114eqtrd 2793 . . 3 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼0 ))
116115ex 416 . 2 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
11711gsumz 18071 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥𝐼0 )) = 0 )
11846, 22, 117syl2anc 587 . . 3 (𝜑 → (𝐺 Σg (𝑥𝐼0 )) = 0 )
119 oveq2 7163 . . . 4 (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐼0 )))
120119eqeq1d 2760 . . 3 (𝐹 = (𝑥𝐼0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥𝐼0 )) = 0 ))
121118, 120syl5ibrcom 250 . 2 (𝜑 → (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = 0 ))
122116, 121impbid 215 1 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  {crab 3074  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ifcif 4423  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   class class class wbr 5035   ↦ cmpt 5115  dom cdm 5527  ran crn 5528   “ cima 5530   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155   ∘f cof 7408  Xcixp 8484   finSupp cfsupp 8871  Basecbs 16546  0gc0g 16776   Σg cgsu 16777  Moorecmre 16916  mrClscmrc 16917  ACScacs 16919  Mndcmnd 17982  SubMndcsubmnd 18026  Grpcgrp 18174  -gcsg 18176  SubGrpcsubg 18345  Cntzccntz 18517   DProd cdprd 19188 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 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7410  df-om 7585  df-1st 7698  df-2nd 7699  df-supp 7841  df-tpos 7907  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-er 8304  df-map 8423  df-ixp 8485  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-fsupp 8872  df-oi 9012  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-n0 11940  df-z 12026  df-uz 12288  df-fz 12945  df-fzo 13088  df-seq 13424  df-hash 13746  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-ress 16554  df-plusg 16641  df-0g 16778  df-gsum 16779  df-mre 16920  df-mrc 16921  df-acs 16923  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-mhm 18027  df-submnd 18028  df-grp 18177  df-minusg 18178  df-sbg 18179  df-mulg 18297  df-subg 18348  df-ghm 18428  df-gim 18471  df-cntz 18519  df-oppg 18546  df-cmn 18980  df-dprd 19190 This theorem is referenced by:  dprdf11  19218
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