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Theorem dprdfeq0 20022
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 2726 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 20012 . . . . . 6 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76feqmptd 6971 . . . . 5 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
87adantr 479 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
91, 2, 3, 4dprdfcl 20013 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
109adantlr 713 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
11 eldprdi.0 . . . . . . . . . . . 12 0 = (0g𝐺)
122ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺dom DProd 𝑆)
133ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → dom 𝑆 = 𝐼)
14 simpr 483 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
15 eqid 2726 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))
1611, 1, 12, 13, 14, 10, 15dprdfid 20017 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥)))
1716simpld 493 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊)
184ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹𝑊)
19 eqid 2726 . . . . . . . . . . . 12 (-g𝐺) = (-g𝐺)
2011, 1, 12, 13, 17, 18, 19dprdfsub 20021 . . . . . . . . . . 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 494 . . . . . . . . . 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 20001 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ V)
2322ad2antrr 724 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐼 ∈ V)
24 fvex 6914 . . . . . . . . . . . . . 14 (𝐹𝑥) ∈ V
2511fvexi 6915 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4583 . . . . . . . . . . . . 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 6916 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ V)
29 eqidd 2727 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))
306ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
3130feqmptd 6971 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹 = (𝑦𝐼 ↦ (𝐹𝑦)))
3223, 27, 28, 29, 31offval2 7710 . . . . . . . . . . 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 7440 . . . . . . . . . 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 494 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥))
35 simplr 767 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg 𝐹) = 0 )
3634, 35oveq12d 7442 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = ((𝐹𝑥)(-g𝐺) 0 ))
37 dprdgrp 20005 . . . . . . . . . . . . 13 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
3812, 37syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Grp)
3930, 14ffvelcdmd 7099 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (Base‘𝐺))
405, 11, 19grpsubid1 19019 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4138, 39, 40syl2anc 582 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4236, 41eqtrd 2766 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = (𝐹𝑥))
4321, 33, 423eqtr3d 2774 . . . . . . . . 9 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))) = (𝐹𝑥))
44 eqid 2726 . . . . . . . . . 10 (Cntz‘𝐺) = (Cntz‘𝐺)
45 grpmnd 18935 . . . . . . . . . . . 12 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
4746ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Mnd)
485subgacs 19155 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49 acsmre 17665 . . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . 14 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
522, 3dprdf2 20007 . . . . . . . . . . . . . . . . 17 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
5352ad2antrr 724 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
5453frnd 6736 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55 mresspw 17605 . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5851, 57sstrid 3991 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59 sspwuni 5108 . . . . . . . . . . . . 13 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
6058, 59sylib 217 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61 eqid 2726 . . . . . . . . . . . . 13 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6261mrccl 17624 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
6350, 60, 62syl2anc 582 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64 subgsubm 19142 . . . . . . . . . . 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 7431 . . . . . . . . . . . . 13 ((𝐹𝑥) = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6766eleq1d 2811 . . . . . . . . . . . 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 7431 . . . . . . . . . . . . 13 ( 0 = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ( 0 (-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6968eleq1d 2811 . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
7170fveq2d 6905 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
7271oveq2d 7440 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = ((𝐹𝑥)(-g𝐺)(𝐹𝑥)))
735, 11, 19grpsubid 19018 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7438, 39, 73syl2anc 582 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7511subg0cl 19128 . . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7877ad2antrr 724 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7972, 78eqeltrd 2826 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8063ad2antrr 724 . . . . . . . . . . . . 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 17638 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8382ad2antrr 724 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
841, 12, 13, 18dprdfcl 20013 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
8584adantr 479 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆𝑦))
8653ffnd 6729 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆 Fn 𝐼)
8786ad2antrr 724 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88 difssd 4132 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89 df-ne 2931 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
90 eldifsn 4795 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦𝐼𝑦𝑥))
9190biimpri 227 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐼𝑦𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9289, 91sylan2br 593 . . . . . . . . . . . . . . . . 17 ((𝑦𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9392adantll 712 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94 fnfvima 7250 . . . . . . . . . . . . . . . 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 4918 . . . . . . . . . . . . . . 15 (((𝐹𝑦) ∈ (𝑆𝑦) ∧ (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9785, 95, 96syl2anc 582 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9883, 97sseldd 3980 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
9919subgsubcl 19131 . . . . . . . . . . . . 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 4571 . . . . . . . . . . 11 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
102101fmpttd 7129 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10320simpld 493 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹) ∈ 𝑊)
10432, 103eqeltrrd 2827 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) ∈ 𝑊)
1051, 12, 13, 104, 44dprdfcntz 20015 . . . . . . . . . 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 20014 . . . . . . . . . 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 19916 . . . . . . . . 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 2827 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10910, 108elind 4195 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
11012, 13, 14, 11, 61dprddisj 20009 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111109, 110eleqtrd 2828 . . . . . 6 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ { 0 })
112 elsni 4650 . . . . . 6 ((𝐹𝑥) ∈ { 0 } → (𝐹𝑥) = 0 )
113111, 112syl 17 . . . . 5 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) = 0 )
114113mpteq2dva 5253 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥𝐼 ↦ (𝐹𝑥)) = (𝑥𝐼0 ))
1158, 114eqtrd 2766 . . 3 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼0 ))
116115ex 411 . 2 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
11711gsumz 18826 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥𝐼0 )) = 0 )
11846, 22, 117syl2anc 582 . . 3 (𝜑 → (𝐺 Σg (𝑥𝐼0 )) = 0 )
119 oveq2 7432 . . . 4 (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐼0 )))
120119eqeq1d 2728 . . 3 (𝐹 = (𝑥𝐼0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥𝐼0 )) = 0 ))
121118, 120syl5ibrcom 246 . 2 (𝜑 → (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = 0 ))
122116, 121impbid 211 1 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  {crab 3419  Vcvv 3462  cdif 3944  cin 3946  wss 3947  ifcif 4533  𝒫 cpw 4607  {csn 4633   cuni 4913   class class class wbr 5153  cmpt 5236  dom cdm 5682  ran crn 5683  cima 5685   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  f cof 7688  Xcixp 8926   finSupp cfsupp 9405  Basecbs 17213  0gc0g 17454   Σg cgsu 17455  Moorecmre 17595  mrClscmrc 17596  ACScacs 17598  Mndcmnd 18727  SubMndcsubmnd 18772  Grpcgrp 18928  -gcsg 18930  SubGrpcsubg 19114  Cntzccntz 19309   DProd cdprd 19993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-0g 17456  df-gsum 17457  df-mre 17599  df-mrc 17600  df-acs 17602  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-subg 19117  df-ghm 19207  df-gim 19253  df-cntz 19311  df-oppg 19340  df-cmn 19780  df-dprd 19995
This theorem is referenced by:  dprdf11  20023
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