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Theorem dprdfeq0 18734
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 2797 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
61, 2, 3, 4, 5dprdff 18724 . . . . . 6 (𝜑𝐹:𝐼⟶(Base‘𝐺))
76feqmptd 6472 . . . . 5 (𝜑𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
87adantr 473 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼 ↦ (𝐹𝑥)))
91, 2, 3, 4dprdfcl 18725 . . . . . . . . 9 ((𝜑𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
109adantlr 707 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (𝑆𝑥))
11 eldprdi.0 . . . . . . . . . . . 12 0 = (0g𝐺)
122ad2antrr 718 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺dom DProd 𝑆)
133ad2antrr 718 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → dom 𝑆 = 𝐼)
14 simpr 478 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
15 eqid 2797 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))
1611, 1, 12, 13, 14, 10, 15dprdfid 18729 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥)))
1716simpld 489 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∈ 𝑊)
184ad2antrr 718 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹𝑊)
19 eqid 2797 . . . . . . . . . . . 12 (-g𝐺) = (-g𝐺)
2011, 1, 12, 13, 17, 18, 19dprdfsub 18733 . . . . . . . . . . 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 490 . . . . . . . . . 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 18713 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ V)
2322ad2antrr 718 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐼 ∈ V)
24 fvex 6422 . . . . . . . . . . . . . 14 (𝐹𝑥) ∈ V
2511fvexi 6423 . . . . . . . . . . . . . 14 0 ∈ V
2624, 25ifex 4323 . . . . . . . . . . . . 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 6424 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ V)
29 eqidd 2798 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) = (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))
306ad2antrr 718 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹:𝐼⟶(Base‘𝐺))
3130feqmptd 6472 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐹 = (𝑦𝐼 ↦ (𝐹𝑦)))
3223, 27, 28, 29, 31offval2 7146 . . . . . . . . . . 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 6892 . . . . . . . . . 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 490 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 ))) = (𝐹𝑥))
35 simplr 786 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg 𝐹) = 0 )
3634, 35oveq12d 6894 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = ((𝐹𝑥)(-g𝐺) 0 ))
37 dprdgrp 18717 . . . . . . . . . . . . 13 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
3812, 37syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Grp)
3930, 14ffvelrnd 6584 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ (Base‘𝐺))
405, 11, 19grpsubid1 17813 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4138, 39, 40syl2anc 580 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺) 0 ) = (𝐹𝑥))
4236, 41eqtrd 2831 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐺 Σg (𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )))(-g𝐺)(𝐺 Σg 𝐹)) = (𝐹𝑥))
4321, 33, 423eqtr3d 2839 . . . . . . . . 9 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐺 Σg (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))) = (𝐹𝑥))
44 eqid 2797 . . . . . . . . . 10 (Cntz‘𝐺) = (Cntz‘𝐺)
45 grpmnd 17742 . . . . . . . . . . . 12 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
462, 37, 453syl 18 . . . . . . . . . . 11 (𝜑𝐺 ∈ Mnd)
4746ad2antrr 718 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝐺 ∈ Mnd)
485subgacs 17939 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
49 acsmre 16624 . . . . . . . . . . . . 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 5692 . . . . . . . . . . . . . 14 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
522, 3dprdf2 18719 . . . . . . . . . . . . . . . . 17 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
5352ad2antrr 718 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺))
5453frnd 6261 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺))
55 mresspw 16564 . . . . . . . . . . . . . . . 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 3806 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5851, 57syl5ss 3807 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
59 sspwuni 4800 . . . . . . . . . . . . 13 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
6058, 59sylib 210 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
61 eqid 2797 . . . . . . . . . . . . 13 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
6261mrccl 16583 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
6350, 60, 62syl2anc 580 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
64 subgsubm 17926 . . . . . . . . . . 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 6883 . . . . . . . . . . . . 13 ((𝐹𝑥) = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6766eleq1d 2861 . . . . . . . . . . . 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 6883 . . . . . . . . . . . . 13 ( 0 = if(𝑦 = 𝑥, (𝐹𝑥), 0 ) → ( 0 (-g𝐺)(𝐹𝑦)) = (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)))
6968eleq1d 2861 . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
7170fveq2d 6413 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
7271oveq2d 6892 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) = ((𝐹𝑥)(-g𝐺)(𝐹𝑥)))
735, 11, 19grpsubid 17812 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐺)) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7438, 39, 73syl2anc 580 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) = 0 )
7511subg0cl 17912 . . . . . . . . . . . . . . . 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 2876 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7877ad2antrr 718 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
7972, 78eqeltrd 2876 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ 𝑦 = 𝑥) → ((𝐹𝑥)(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8063ad2antrr 718 . . . . . . . . . . . . 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 16597 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
8382ad2antrr 718 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
841, 12, 13, 18dprdfcl 18725 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (𝐹𝑦) ∈ (𝑆𝑦))
8584adantr 473 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆𝑦))
8653ffnd 6255 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → 𝑆 Fn 𝐼)
8786ad2antrr 718 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼)
88 difssd 3934 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼)
89 df-ne 2970 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 ↔ ¬ 𝑦 = 𝑥)
90 eldifsn 4504 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦𝐼𝑦𝑥))
9190biimpri 220 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐼𝑦𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9289, 91sylan2br 589 . . . . . . . . . . . . . . . . 17 ((𝑦𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
9392adantll 706 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥}))
94 fnfvima 6723 . . . . . . . . . . . . . . . 16 ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9587, 88, 93, 94syl3anc 1491 . . . . . . . . . . . . . . 15 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
96 elunii 4631 . . . . . . . . . . . . . . 15 (((𝐹𝑦) ∈ (𝑆𝑦) ∧ (𝑆𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9785, 95, 96syl2anc 580 . . . . . . . . . . . . . 14 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥})))
9883, 97sseldd 3797 . . . . . . . . . . . . 13 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
9919subgsubcl 17915 . . . . . . . . . . . . 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 1491 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) ∧ ¬ 𝑦 = 𝑥) → ( 0 (-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10167, 69, 79, 100ifbothda 4312 . . . . . . . . . . 11 ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) ∧ 𝑦𝐼) → (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
102101fmpttd 6609 . . . . . . . . . 10 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10320simpld 489 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ if(𝑦 = 𝑥, (𝐹𝑥), 0 )) ∘𝑓 (-g𝐺)𝐹) ∈ 𝑊)
10432, 103eqeltrrd 2877 . . . . . . . . . . 11 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝑦𝐼 ↦ (if(𝑦 = 𝑥, (𝐹𝑥), 0 )(-g𝐺)(𝐹𝑦))) ∈ 𝑊)
1051, 12, 13, 104, 44dprdfcntz 18727 . . . . . . . . . 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 18726 . . . . . . . . . 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 18630 . . . . . . . . 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 2877 . . . . . . . 8 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
10910, 108elind 3994 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
11012, 13, 14, 11, 61dprddisj 18721 . . . . . . 7 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = { 0 })
111109, 110eleqtrd 2878 . . . . . 6 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) ∈ { 0 })
112 elsni 4383 . . . . . 6 ((𝐹𝑥) ∈ { 0 } → (𝐹𝑥) = 0 )
113111, 112syl 17 . . . . 5 (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥𝐼) → (𝐹𝑥) = 0 )
114113mpteq2dva 4935 . . . 4 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥𝐼 ↦ (𝐹𝑥)) = (𝑥𝐼0 ))
1158, 114eqtrd 2831 . . 3 ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥𝐼0 ))
116115ex 402 . 2 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
11711gsumz 17686 . . . 4 ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg (𝑥𝐼0 )) = 0 )
11846, 22, 117syl2anc 580 . . 3 (𝜑 → (𝐺 Σg (𝑥𝐼0 )) = 0 )
119 oveq2 6884 . . . 4 (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝐼0 )))
120119eqeq1d 2799 . . 3 (𝐹 = (𝑥𝐼0 ) → ((𝐺 Σg 𝐹) = 0 ↔ (𝐺 Σg (𝑥𝐼0 )) = 0 ))
121118, 120syl5ibrcom 239 . 2 (𝜑 → (𝐹 = (𝑥𝐼0 ) → (𝐺 Σg 𝐹) = 0 ))
122116, 121impbid 204 1 (𝜑 → ((𝐺 Σg 𝐹) = 0𝐹 = (𝑥𝐼0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wne 2969  {crab 3091  Vcvv 3383  cdif 3764  cin 3766  wss 3767  ifcif 4275  𝒫 cpw 4347  {csn 4366   cuni 4626   class class class wbr 4841  cmpt 4920  dom cdm 5310  ran crn 5311  cima 5313   Fn wfn 6094  wf 6095  cfv 6099  (class class class)co 6876  𝑓 cof 7127  Xcixp 8146   finSupp cfsupp 8515  Basecbs 16181  0gc0g 16412   Σg cgsu 16413  Moorecmre 16554  mrClscmrc 16555  ACScacs 16557  Mndcmnd 17606  SubMndcsubmnd 17646  Grpcgrp 17735  -gcsg 17737  SubGrpcsubg 17898  Cntzccntz 18057   DProd cdprd 18705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-inf2 8786  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-se 5270  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-of 7129  df-om 7298  df-1st 7399  df-2nd 7400  df-supp 7531  df-tpos 7588  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-er 7980  df-map 8095  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-fsupp 8516  df-oi 8655  df-card 9049  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-n0 11577  df-z 11663  df-uz 11927  df-fz 12577  df-fzo 12717  df-seq 13052  df-hash 13367  df-ndx 16184  df-slot 16185  df-base 16187  df-sets 16188  df-ress 16189  df-plusg 16277  df-0g 16414  df-gsum 16415  df-mre 16558  df-mrc 16559  df-acs 16561  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-mhm 17647  df-submnd 17648  df-grp 17738  df-minusg 17739  df-sbg 17740  df-mulg 17854  df-subg 17901  df-ghm 17968  df-gim 18011  df-cntz 18059  df-oppg 18085  df-cmn 18507  df-dprd 18707
This theorem is referenced by:  dprdf11  18735
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