| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eldprdi.w | . . . . . . 7
⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | 
| 2 |  | eldprdi.1 | . . . . . . 7
⊢ (𝜑 → 𝐺dom DProd 𝑆) | 
| 3 |  | eldprdi.2 | . . . . . . 7
⊢ (𝜑 → dom 𝑆 = 𝐼) | 
| 4 |  | eldprdi.3 | . . . . . . 7
⊢ (𝜑 → 𝐹 ∈ 𝑊) | 
| 5 |  | eqid 2736 | . . . . . . 7
⊢
(Base‘𝐺) =
(Base‘𝐺) | 
| 6 | 1, 2, 3, 4, 5 | dprdff 20033 | . . . . . 6
⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) | 
| 7 | 6 | feqmptd 6976 | . . . . 5
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) | 
| 8 | 7 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥))) | 
| 9 | 1, 2, 3, 4 | dprdfcl 20034 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥)) | 
| 10 | 9 | adantlr 715 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (𝑆‘𝑥)) | 
| 11 |  | eldprdi.0 | . . . . . . . . . . . 12
⊢  0 =
(0g‘𝐺) | 
| 12 | 2 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆) | 
| 13 | 3 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼) | 
| 14 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | 
| 15 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) | 
| 16 | 11, 1, 12, 13, 14, 10, 15 | dprdfid 20038 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊 ∧ (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥))) | 
| 17 | 16 | simpld 494 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∈ 𝑊) | 
| 18 | 4 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝑊) | 
| 19 |  | eqid 2736 | . . . . . . . . . . . 12
⊢
(-g‘𝐺) = (-g‘𝐺) | 
| 20 | 11, 1, 12, 13, 17, 18, 19 | dprdfsub 20042 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f
(-g‘𝐺)𝐹) ∈ 𝑊 ∧ (𝐺 Σg ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f
(-g‘𝐺)𝐹)) = ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)))(-g‘𝐺)(𝐺 Σg 𝐹)))) | 
| 21 | 20 | simprd 495 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f
(-g‘𝐺)𝐹)) = ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)))(-g‘𝐺)(𝐺 Σg 𝐹))) | 
| 22 | 2, 3 | dprddomcld 20022 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ V) | 
| 23 | 22 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V) | 
| 24 |  | fvex 6918 | . . . . . . . . . . . . . 14
⊢ (𝐹‘𝑥) ∈ V | 
| 25 | 11 | fvexi 6919 | . . . . . . . . . . . . . 14
⊢  0 ∈
V | 
| 26 | 24, 25 | ifex 4575 | . . . . . . . . . . . . 13
⊢ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) ∈
V | 
| 27 | 26 | a1i 11 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) ∈
V) | 
| 28 |  | fvexd 6920 | . . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ V) | 
| 29 |  | eqidd 2737 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) | 
| 30 | 6 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹:𝐼⟶(Base‘𝐺)) | 
| 31 | 30 | feqmptd 6976 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐹 = (𝑦 ∈ 𝐼 ↦ (𝐹‘𝑦))) | 
| 32 | 23, 27, 28, 29, 31 | offval2 7718 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f
(-g‘𝐺)𝐹) = (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦)))) | 
| 33 | 32 | oveq2d 7448 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f
(-g‘𝐺)𝐹)) = (𝐺 Σg (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))))) | 
| 34 | 16 | simprd 495 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ))) = (𝐹‘𝑥)) | 
| 35 |  | simplr 768 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝐹) = 0 ) | 
| 36 | 34, 35 | oveq12d 7450 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)))(-g‘𝐺)(𝐺 Σg 𝐹)) = ((𝐹‘𝑥)(-g‘𝐺) 0 )) | 
| 37 |  | dprdgrp 20026 | . . . . . . . . . . . . 13
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | 
| 38 | 12, 37 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Grp) | 
| 39 | 30, 14 | ffvelcdmd 7104 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘𝐺)) | 
| 40 | 5, 11, 19 | grpsubid1 19044 | . . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥)) | 
| 41 | 38, 39, 40 | syl2anc 584 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺) 0 ) = (𝐹‘𝑥)) | 
| 42 | 36, 41 | eqtrd 2776 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σg (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)))(-g‘𝐺)(𝐺 Σg 𝐹)) = (𝐹‘𝑥)) | 
| 43 | 21, 33, 42 | 3eqtr3d 2784 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦)))) = (𝐹‘𝑥)) | 
| 44 |  | eqid 2736 | . . . . . . . . . 10
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) | 
| 45 |  | grpmnd 18959 | . . . . . . . . . . . 12
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | 
| 46 | 2, 37, 45 | 3syl 18 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ Mnd) | 
| 47 | 46 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd) | 
| 48 | 5 | subgacs 19180 | . . . . . . . . . . . . 13
⊢ (𝐺 ∈ Grp →
(SubGrp‘𝐺) ∈
(ACS‘(Base‘𝐺))) | 
| 49 |  | acsmre 17696 | . . . . . . . . . . . . 13
⊢
((SubGrp‘𝐺)
∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) | 
| 50 | 38, 48, 49 | 3syl 18 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) | 
| 51 |  | imassrn 6088 | . . . . . . . . . . . . . 14
⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆 | 
| 52 | 2, 3 | dprdf2 20028 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) | 
| 53 | 52 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺)) | 
| 54 | 53 | frnd 6743 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ (SubGrp‘𝐺)) | 
| 55 |  | mresspw 17636 | . . . . . . . . . . . . . . . 16
⊢
((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)) | 
| 56 | 50, 55 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)) | 
| 57 | 54, 56 | sstrd 3993 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺)) | 
| 58 | 51, 57 | sstrid 3994 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺)) | 
| 59 |  | sspwuni 5099 | . . . . . . . . . . . . 13
⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆
“ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) | 
| 60 | 58, 59 | sylib 218 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) | 
| 61 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢
(mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | 
| 62 | 61 | mrccl 17655 | . . . . . . . . . . . 12
⊢
(((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) | 
| 63 | 50, 60, 62 | syl2anc 584 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((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)) | 
| 65 | 63, 64 | syl 17 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ∈ (SubMnd‘𝐺)) | 
| 66 |  | oveq1 7439 | . . . . . . . . . . . . 13
⊢ ((𝐹‘𝑥) = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) = (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))) | 
| 67 | 66 | eleq1d 2825 | . . . . . . . . . . . 12
⊢ ((𝐹‘𝑥) = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → (((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ↔ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))))) | 
| 68 |  | oveq1 7439 | . . . . . . . . . . . . 13
⊢ ( 0 = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → ( 0
(-g‘𝐺)(𝐹‘𝑦)) = (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))) | 
| 69 | 68 | eleq1d 2825 | . . . . . . . . . . . 12
⊢ ( 0 = if(𝑦 = 𝑥, (𝐹‘𝑥), 0 ) → (( 0
(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ↔ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))))) | 
| 70 |  | simpr 484 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥) | 
| 71 | 70 | fveq2d 6909 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → (𝐹‘𝑦) = (𝐹‘𝑥)) | 
| 72 | 71 | oveq2d 7448 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) = ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥))) | 
| 73 | 5, 11, 19 | grpsubid 19043 | . . . . . . . . . . . . . . . 16
⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐺)) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 ) | 
| 74 | 38, 39, 73 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) = 0 ) | 
| 75 | 11 | subg0cl 19153 | . . . . . . . . . . . . . . . 16
⊢
(((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → 0 ∈
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) | 
| 76 | 63, 75 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 0 ∈
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) | 
| 77 | 74, 76 | eqeltrd 2840 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 78 | 77 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑥)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 79 | 72, 78 | eqeltrd 2840 | . . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ 𝑦 = 𝑥) → ((𝐹‘𝑥)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 80 | 63 | ad2antrr 726 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) | 
| 81 | 80, 75 | syl 17 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 0 ∈
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) | 
| 82 | 50, 61, 60 | mrcssidd 17669 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 83 | 82 | ad2antrr 726 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 84 | 1, 12, 13, 18 | dprdfcl 20034 | . . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (𝐹‘𝑦) ∈ (𝑆‘𝑦)) | 
| 85 | 84 | adantr 480 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ (𝑆‘𝑦)) | 
| 86 | 53 | ffnd 6736 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑆 Fn 𝐼) | 
| 87 | 86 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑆 Fn 𝐼) | 
| 88 |  | difssd 4136 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐼 ∖ {𝑥}) ⊆ 𝐼) | 
| 89 |  | df-ne 2940 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥) | 
| 90 |  | eldifsn 4785 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ (𝐼 ∖ {𝑥}) ↔ (𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥)) | 
| 91 | 90 | biimpri 228 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑦 ∈ 𝐼 ∧ 𝑦 ≠ 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥})) | 
| 92 | 89, 91 | sylan2br 595 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥})) | 
| 93 | 92 | adantll 714 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → 𝑦 ∈ (𝐼 ∖ {𝑥})) | 
| 94 |  | fnfvima 7254 | . . . . . . . . . . . . . . . 16
⊢ ((𝑆 Fn 𝐼 ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼 ∧ 𝑦 ∈ (𝐼 ∖ {𝑥})) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) | 
| 95 | 87, 88, 93, 94 | syl3anc 1372 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) | 
| 96 |  | elunii 4911 | . . . . . . . . . . . . . . 15
⊢ (((𝐹‘𝑦) ∈ (𝑆‘𝑦) ∧ (𝑆‘𝑦) ∈ (𝑆 “ (𝐼 ∖ {𝑥}))) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥}))) | 
| 97 | 85, 95, 96 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ∪ (𝑆 “ (𝐼 ∖ {𝑥}))) | 
| 98 | 83, 97 | sseldd 3983 | . . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 99 | 19 | subgsubcl 19156 | . . . . . . . . . . . . 13
⊢
((((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ 0 ∈
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) ∧ (𝐹‘𝑦) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) → ( 0
(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 100 | 80, 81, 98, 99 | syl3anc 1372 | . . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝐺 Σg
𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) ∧ ¬ 𝑦 = 𝑥) → ( 0 (-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 101 | 67, 69, 79, 100 | ifbothda 4563 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) ∧ 𝑦 ∈ 𝐼) → (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦)) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 102 | 101 | fmpttd 7134 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))):𝐼⟶((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 103 | 20 | simpld 494 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, (𝐹‘𝑥), 0 )) ∘f
(-g‘𝐺)𝐹) ∈ 𝑊) | 
| 104 | 32, 103 | eqeltrrd 2841 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))) ∈ 𝑊) | 
| 105 | 1, 12, 13, 104, 44 | dprdfcntz 20036 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ran (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))) ⊆ ((Cntz‘𝐺)‘ran (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))))) | 
| 106 | 1, 12, 13, 104 | dprdffsupp 20035 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦))) finSupp 0 ) | 
| 107 | 11, 44, 47, 23, 65, 102, 105, 106 | gsumzsubmcl 19937 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑦 ∈ 𝐼 ↦ (if(𝑦 = 𝑥, (𝐹‘𝑥), 0
)(-g‘𝐺)(𝐹‘𝑦)))) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 108 | 43, 107 | eqeltrrd 2841 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) | 
| 109 | 10, 108 | elind 4199 | . . . . . . 7
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))))) | 
| 110 | 12, 13, 14, 11, 61 | dprddisj 20030 | . . . . . . 7
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) = { 0 }) | 
| 111 | 109, 110 | eleqtrd 2842 | . . . . . 6
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ { 0 }) | 
| 112 |  | elsni 4642 | . . . . . 6
⊢ ((𝐹‘𝑥) ∈ { 0 } → (𝐹‘𝑥) = 0 ) | 
| 113 | 111, 112 | syl 17 | . . . . 5
⊢ (((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = 0 ) | 
| 114 | 113 | mpteq2dva 5241 | . . . 4
⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐼 ↦ 0 )) | 
| 115 | 8, 114 | eqtrd 2776 | . . 3
⊢ ((𝜑 ∧ (𝐺 Σg 𝐹) = 0 ) → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 )) | 
| 116 | 115 | ex 412 | . 2
⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 → 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))) | 
| 117 | 11 | gsumz 18850 | . . . 4
⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ V) → (𝐺 Σg
(𝑥 ∈ 𝐼 ↦ 0 )) = 0 ) | 
| 118 | 46, 22, 117 | syl2anc 584 | . . 3
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 ) | 
| 119 |  | oveq2 7440 | . . . 4
⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg
𝐹) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 ))) | 
| 120 | 119 | eqeq1d 2738 | . . 3
⊢ (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → ((𝐺 Σg
𝐹) = 0 ↔ (𝐺 Σg (𝑥 ∈ 𝐼 ↦ 0 )) = 0 )) | 
| 121 | 118, 120 | syl5ibrcom 247 | . 2
⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ) → (𝐺 Σg
𝐹) = 0 )) | 
| 122 | 116, 121 | impbid 212 | 1
⊢ (𝜑 → ((𝐺 Σg 𝐹) = 0 ↔ 𝐹 = (𝑥 ∈ 𝐼 ↦ 0 ))) |