MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdfeq0 Structured version   Visualization version   GIF version

Theorem dprdfeq0 20043
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 2736 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 20033 . . . . . 6 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76feqmptd 6976 . . . . 5 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
87adantr 480 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
91, 2, 3, 4dprdfcl 20034 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
109adantlr 715 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
11 eldprdi.0 . . . . . . . . . . . 12 0 = (0g𝐺)
122ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺dom DProd 𝑆)
133ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → dom 𝑆 = 𝐼)
14 simpr 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
15 eqid 2736 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))
1611, 1, 12, 13, 14, 10, 15dprdfid 20038 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥)))
1716simpld 494 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊)
184ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹𝑊)
19 eqid 2736 . . . . . . . . . . . 12 (-g𝐺) = (-g𝐺)
2011, 1, 12, 13, 17, 18, 19dprdfsub 20042 . . . . . . . . . . 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 495 . . . . . . . . . 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 20022 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ V)
2322ad2antrr 726 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐼 ∈ V)
24 fvex 6918 . . . . . . . . . . . . . 14 (𝐹𝑥) ∈ V
2511fvexi 6919 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4575 . . . . . . . . . . . . 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 6920 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ V)
29 eqidd 2737 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))
306ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
3130feqmptd 6976 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹 = (𝑦𝐼 ↦ (𝐹𝑦)))
3223, 27, 28, 29, 31offval2 7718 . . . . . . . . . . 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 7448 . . . . . . . . . 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 495 . . . . . . . . . . . 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 7450 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = ((𝐹𝑥)(-g𝐺) 0 ))
37 dprdgrp 20026 . . . . . . . . . . . . 13 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
3812, 37syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Grp)
3930, 14ffvelcdmd 7104 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (Base‘𝐺))
405, 11, 19grpsubid1 19044 . . . . . . . . . . . 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 2776 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = (𝐹𝑥))
4321, 33, 423eqtr3d 2784 . . . . . . . . 9 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))) = (𝐹𝑥))
44 eqid 2736 . . . . . . . . . 10 (Cntz‘𝐺) = (Cntz‘𝐺)
45 grpmnd 18959 . . . . . . . . . . . 12 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
4746ad2antrr 726 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Mnd)
485subgacs 19180 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49 acsmre 17696 . . . . . . . . . . . . 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 6088 . . . . . . . . . . . . . 14 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
522, 3dprdf2 20028 . . . . . . . . . . . . . . . . 17 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
5352ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
5453frnd 6743 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55 mresspw 17636 . . . . . . . . . . . . . . . 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 3993 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5851, 57sstrid 3994 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59 sspwuni 5099 . . . . . . . . . . . . 13 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
6058, 59sylib 218 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61 eqid 2736 . . . . . . . . . . . . 13 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6261mrccl 17655 . . . . . . . . . . . 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 19167 . . . . . . . . . . 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 7439 . . . . . . . . . . . . 13 ((𝐹𝑥) = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6766eleq1d 2825 . . . . . . . . . . . 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 7439 . . . . . . . . . . . . 13 ( 0 = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ( 0 (-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6968eleq1d 2825 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
7170fveq2d 6909 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
7271oveq2d 7448 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = ((𝐹𝑥)(-g𝐺)(𝐹𝑥)))
735, 11, 19grpsubid 19043 . . . . . . . . . . . . . . . 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 19153 . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7877ad2antrr 726 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7972, 78eqeltrd 2840 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8063ad2antrr 726 . . . . . . . . . . . . 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 17669 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8382ad2antrr 726 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
841, 12, 13, 18dprdfcl 20034 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
8584adantr 480 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆𝑦))
8653ffnd 6736 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆 Fn 𝐼)
8786ad2antrr 726 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88 difssd 4136 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89 df-ne 2940 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
90 eldifsn 4785 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦𝐼𝑦𝑥))
9190biimpri 228 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐼𝑦𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9289, 91sylan2br 595 . . . . . . . . . . . . . . . . 17 ((𝑦𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9392adantll 714 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94 fnfvima 7254 . . . . . . . . . . . . . . . 16 ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9587, 88, 93, 94syl3anc 1372 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96 elunii 4911 . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
9919subgsubcl 19156 . . . . . . . . . . . . 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 1372 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → ( 0 (-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10167, 69, 79, 100ifbothda 4563 . . . . . . . . . . 11 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
102101fmpttd 7134 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10320simpld 494 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘f (-g𝐺)𝐹) ∈ 𝑊)
10432, 103eqeltrrd 2841 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) ∈ 𝑊)
1051, 12, 13, 104, 44dprdfcntz 20036 . . . . . . . . . 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 20035 . . . . . . . . . 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 19937 . . . . . . . . 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 2841 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10910, 108elind 4199 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
11012, 13, 14, 11, 61dprddisj 20030 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111109, 110eleqtrd 2842 . . . . . 6 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ { 0 })
112 elsni 4642 . . . . . 6 ((𝐹𝑥) ∈ { 0 } → (𝐹𝑥) = 0 )
113111, 112syl 17 . . . . 5 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) = 0 )
114113mpteq2dva 5241 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥𝐼 ↦ (𝐹𝑥)) = (𝑥𝐼0 ))
1158, 114eqtrd 2776 . . 3 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼0 ))
116115ex 412 . 2 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
11711gsumz 18850 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥𝐼0 )) = 0 )
11846, 22, 117syl2anc 584 . . 3 (𝜑 → (𝐺 Σg (𝑥𝐼0 )) = 0 )
119 oveq2 7440 . . . 4 (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐼0 )))
120119eqeq1d 2738 . . 3 (𝐹 = (𝑥𝐼0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥𝐼0 )) = 0 ))
121118, 120syl5ibrcom 247 . 2 (𝜑 → (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = 0 ))
122116, 121impbid 212 1 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  {crab 3435  Vcvv 3479  cdif 3947  cin 3949  wss 3950  ifcif 4524  𝒫 cpw 4599  {csn 4625   cuni 4906   class class class wbr 5142  cmpt 5224  dom cdm 5684  ran crn 5685  cima 5687   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  f cof 7696  Xcixp 8938   finSupp cfsupp 9402  Basecbs 17248  0gc0g 17485   Σg cgsu 17486  Moorecmre 17626  mrClscmrc 17627  ACScacs 17629  Mndcmnd 18748  SubMndcsubmnd 18796  Grpcgrp 18952  -gcsg 18954  SubGrpcsubg 19139  Cntzccntz 19334   DProd cdprd 20014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-gim 19278  df-cntz 19336  df-oppg 19365  df-cmn 19801  df-dprd 20016
This theorem is referenced by:  dprdf11  20044
  Copyright terms: Public domain W3C validator