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Theorem dprdfeq0 19073
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 2818 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 19063 . . . . . 6 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76feqmptd 6726 . . . . 5 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
87adantr 481 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
91, 2, 3, 4dprdfcl 19064 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
109adantlr 711 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
11 eldprdi.0 . . . . . . . . . . . 12 0 = (0g𝐺)
122ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺dom DProd 𝑆)
133ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → dom 𝑆 = 𝐼)
14 simpr 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
15 eqid 2818 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))
1611, 1, 12, 13, 14, 10, 15dprdfid 19068 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥)))
1716simpld 495 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊)
184ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹𝑊)
19 eqid 2818 . . . . . . . . . . . 12 (-g𝐺) = (-g𝐺)
2011, 1, 12, 13, 17, 18, 19dprdfsub 19072 . . . . . . . . . . 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 496 . . . . . . . . . 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 19052 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ V)
2322ad2antrr 722 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐼 ∈ V)
24 fvex 6676 . . . . . . . . . . . . . 14 (𝐹𝑥) ∈ V
2511fvexi 6677 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4511 . . . . . . . . . . . . 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 6678 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ V)
29 eqidd 2819 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))
306ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
3130feqmptd 6726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹 = (𝑦𝐼 ↦ (𝐹𝑦)))
3223, 27, 28, 29, 31offval2 7415 . . . . . . . . . . 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 7161 . . . . . . . . . 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 496 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥))
35 simplr 765 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg 𝐹) = 0 )
3634, 35oveq12d 7163 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = ((𝐹𝑥)(-g𝐺) 0 ))
37 dprdgrp 19056 . . . . . . . . . . . . 13 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
3812, 37syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Grp)
3930, 14ffvelrnd 6844 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (Base‘𝐺))
405, 11, 19grpsubid1 18122 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4138, 39, 40syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4236, 41eqtrd 2853 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = (𝐹𝑥))
4321, 33, 423eqtr3d 2861 . . . . . . . . 9 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))) = (𝐹𝑥))
44 eqid 2818 . . . . . . . . . 10 (Cntz‘𝐺) = (Cntz‘𝐺)
45 grpmnd 18048 . . . . . . . . . . . 12 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
4746ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Mnd)
485subgacs 18251 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49 acsmre 16911 . . . . . . . . . . . . 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 5933 . . . . . . . . . . . . . 14 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
522, 3dprdf2 19058 . . . . . . . . . . . . . . . . 17 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
5352ad2antrr 722 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
5453frnd 6514 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55 mresspw 16851 . . . . . . . . . . . . . . . 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 3974 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5851, 57sstrid 3975 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59 sspwuni 5013 . . . . . . . . . . . . 13 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
6058, 59sylib 219 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61 eqid 2818 . . . . . . . . . . . . 13 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6261mrccl 16870 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
6350, 60, 62syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64 subgsubm 18239 . . . . . . . . . . 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 7152 . . . . . . . . . . . . 13 ((𝐹𝑥) = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6766eleq1d 2894 . . . . . . . . . . . 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 7152 . . . . . . . . . . . . 13 ( 0 = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ( 0 (-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6968eleq1d 2894 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
7170fveq2d 6667 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
7271oveq2d 7161 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = ((𝐹𝑥)(-g𝐺)(𝐹𝑥)))
735, 11, 19grpsubid 18121 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7438, 39, 73syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7511subg0cl 18225 . . . . . . . . . . . . . . . 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 2910 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7877ad2antrr 722 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7972, 78eqeltrd 2910 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8063ad2antrr 722 . . . . . . . . . . . . 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 16884 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8382ad2antrr 722 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
841, 12, 13, 18dprdfcl 19064 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
8584adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆𝑦))
8653ffnd 6508 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆 Fn 𝐼)
8786ad2antrr 722 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88 difssd 4106 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89 df-ne 3014 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
90 eldifsn 4711 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦𝐼𝑦𝑥))
9190biimpri 229 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐼𝑦𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9289, 91sylan2br 594 . . . . . . . . . . . . . . . . 17 ((𝑦𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9392adantll 710 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94 fnfvima 6986 . . . . . . . . . . . . . . . 16 ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9587, 88, 93, 94syl3anc 1363 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96 elunii 4835 . . . . . . . . . . . . . . 15 (((𝐹𝑦) ∈ (𝑆𝑦) ∧ (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9785, 95, 96syl2anc 584 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9883, 97sseldd 3965 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
9919subgsubcl 18228 . . . . . . . . . . . . 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 1363 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → ( 0 (-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10167, 69, 79, 100ifbothda 4500 . . . . . . . . . . 11 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
102101fmpttd 6871 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10320simpld 495 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹) ∈ 𝑊)
10432, 103eqeltrrd 2911 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) ∈ 𝑊)
1051, 12, 13, 104, 44dprdfcntz 19066 . . . . . . . . . 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 19065 . . . . . . . . . 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 18967 . . . . . . . . 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 2911 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10910, 108elind 4168 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
11012, 13, 14, 11, 61dprddisj 19060 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111109, 110eleqtrd 2912 . . . . . 6 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ { 0 })
112 elsni 4574 . . . . . 6 ((𝐹𝑥) ∈ { 0 } → (𝐹𝑥) = 0 )
113111, 112syl 17 . . . . 5 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) = 0 )
114113mpteq2dva 5152 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥𝐼 ↦ (𝐹𝑥)) = (𝑥𝐼0 ))
1158, 114eqtrd 2853 . . 3 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼0 ))
116115ex 413 . 2 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
11711gsumz 17988 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥𝐼0 )) = 0 )
11846, 22, 117syl2anc 584 . . 3 (𝜑 → (𝐺 Σg (𝑥𝐼0 )) = 0 )
119 oveq2 7153 . . . 4 (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐼0 )))
120119eqeq1d 2820 . . 3 (𝐹 = (𝑥𝐼0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥𝐼0 )) = 0 ))
121118, 120syl5ibrcom 248 . 2 (𝜑 → (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = 0 ))
122116, 121impbid 213 1 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  {crab 3139  Vcvv 3492  cdif 3930  cin 3932  wss 3933  ifcif 4463  𝒫 cpw 4535  {csn 4557   cuni 4830   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396  Xcixp 8449   finSupp cfsupp 8821  Basecbs 16471  0gc0g 16701   Σg cgsu 16702  Moorecmre 16841  mrClscmrc 16842  ACScacs 16844  Mndcmnd 17899  SubMndcsubmnd 17943  Grpcgrp 18041  -gcsg 18043  SubGrpcsubg 18211  Cntzccntz 18383   DProd cdprd 19044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-ghm 18294  df-gim 18337  df-cntz 18385  df-oppg 18412  df-cmn 18837  df-dprd 19046
This theorem is referenced by:  dprdf11  19074
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