| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . . 3
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) | 
| 2 |  | eqid 2736 | . . 3
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 3 |  | eqid 2736 | . . 3
⊢
(mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) | 
| 4 |  | dprdf1o.1 | . . . 4
⊢ (𝜑 → 𝐺dom DProd 𝑆) | 
| 5 |  | dprdgrp 20026 | . . . 4
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | 
| 6 | 4, 5 | syl 17 | . . 3
⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 7 |  | dprdf1o.3 | . . . . 5
⊢ (𝜑 → 𝐹:𝐽–1-1-onto→𝐼) | 
| 8 |  | f1of1 6846 | . . . . 5
⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–1-1→𝐼) | 
| 9 | 7, 8 | syl 17 | . . . 4
⊢ (𝜑 → 𝐹:𝐽–1-1→𝐼) | 
| 10 |  | dprdf1o.2 | . . . . 5
⊢ (𝜑 → dom 𝑆 = 𝐼) | 
| 11 | 4, 10 | dprddomcld 20022 | . . . 4
⊢ (𝜑 → 𝐼 ∈ V) | 
| 12 |  | f1dmex 7982 | . . . 4
⊢ ((𝐹:𝐽–1-1→𝐼 ∧ 𝐼 ∈ V) → 𝐽 ∈ V) | 
| 13 | 9, 11, 12 | syl2anc 584 | . . 3
⊢ (𝜑 → 𝐽 ∈ V) | 
| 14 | 4, 10 | dprdf2 20028 | . . . 4
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) | 
| 15 |  | f1of 6847 | . . . . 5
⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽⟶𝐼) | 
| 16 | 7, 15 | syl 17 | . . . 4
⊢ (𝜑 → 𝐹:𝐽⟶𝐼) | 
| 17 |  | fco 6759 | . . . 4
⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ 𝐹:𝐽⟶𝐼) → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺)) | 
| 18 | 14, 16, 17 | syl2anc 584 | . . 3
⊢ (𝜑 → (𝑆 ∘ 𝐹):𝐽⟶(SubGrp‘𝐺)) | 
| 19 | 4 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐺dom DProd 𝑆) | 
| 20 | 10 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → dom 𝑆 = 𝐼) | 
| 21 | 16 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽⟶𝐼) | 
| 22 |  | simpr1 1194 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐽) | 
| 23 | 21, 22 | ffvelcdmd 7104 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ∈ 𝐼) | 
| 24 |  | simpr2 1195 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐽) | 
| 25 | 21, 24 | ffvelcdmd 7104 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑦) ∈ 𝐼) | 
| 26 |  | simpr3 1196 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | 
| 27 | 9 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → 𝐹:𝐽–1-1→𝐼) | 
| 28 |  | f1fveq 7283 | . . . . . . . 8
⊢ ((𝐹:𝐽–1-1→𝐼 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦)) | 
| 29 | 27, 22, 24, 28 | syl12anc 836 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝑥 = 𝑦)) | 
| 30 | 29 | necon3bid 2984 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝐹‘𝑥) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ 𝑦)) | 
| 31 | 26, 30 | mpbird 257 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝐹‘𝑥) ≠ (𝐹‘𝑦)) | 
| 32 | 19, 20, 23, 25, 31, 1 | dprdcntz 20029 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → (𝑆‘(𝐹‘𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦)))) | 
| 33 |  | fvco3 7007 | . . . . 5
⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥))) | 
| 34 | 21, 22, 33 | syl2anc 584 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥))) | 
| 35 |  | fvco3 7007 | . . . . . 6
⊢ ((𝐹:𝐽⟶𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦))) | 
| 36 | 21, 24, 35 | syl2anc 584 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑦) = (𝑆‘(𝐹‘𝑦))) | 
| 37 | 36 | fveq2d 6909 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘(𝐹‘𝑦)))) | 
| 38 | 32, 34, 37 | 3sstr4d 4038 | . . 3
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ≠ 𝑦)) → ((𝑆 ∘ 𝐹)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ∘ 𝐹)‘𝑦))) | 
| 39 | 16, 33 | sylan 580 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹)‘𝑥) = (𝑆‘(𝐹‘𝑥))) | 
| 40 |  | imaco 6270 | . . . . . . . . 9
⊢ ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥}))) | 
| 41 | 7 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐹:𝐽–1-1-onto→𝐼) | 
| 42 |  | dff1o3 6853 | . . . . . . . . . . . . 13
⊢ (𝐹:𝐽–1-1-onto→𝐼 ↔ (𝐹:𝐽–onto→𝐼 ∧ Fun ◡𝐹)) | 
| 43 | 42 | simprbi 496 | . . . . . . . . . . . 12
⊢ (𝐹:𝐽–1-1-onto→𝐼 → Fun ◡𝐹) | 
| 44 |  | imadif 6649 | . . . . . . . . . . . 12
⊢ (Fun
◡𝐹 → (𝐹 “ (𝐽 ∖ {𝑥})) = ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥}))) | 
| 45 | 41, 43, 44 | 3syl 18 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥}))) | 
| 46 |  | f1ofo 6854 | . . . . . . . . . . . . 13
⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹:𝐽–onto→𝐼) | 
| 47 |  | foima 6824 | . . . . . . . . . . . . 13
⊢ (𝐹:𝐽–onto→𝐼 → (𝐹 “ 𝐽) = 𝐼) | 
| 48 | 41, 46, 47 | 3syl 18 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝐽) = 𝐼) | 
| 49 |  | f1ofn 6848 | . . . . . . . . . . . . . . 15
⊢ (𝐹:𝐽–1-1-onto→𝐼 → 𝐹 Fn 𝐽) | 
| 50 | 7, 49 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 Fn 𝐽) | 
| 51 |  | fnsnfv 6987 | . . . . . . . . . . . . . 14
⊢ ((𝐹 Fn 𝐽 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥})) | 
| 52 | 50, 51 | sylan 580 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥})) | 
| 53 | 52 | eqcomd 2742 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ {𝑥}) = {(𝐹‘𝑥)}) | 
| 54 | 48, 53 | difeq12d 4126 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝐹 “ 𝐽) ∖ (𝐹 “ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)})) | 
| 55 | 45, 54 | eqtrd 2776 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ (𝐽 ∖ {𝑥})) = (𝐼 ∖ {(𝐹‘𝑥)})) | 
| 56 | 55 | imaeq2d 6077 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝑆 “ (𝐹 “ (𝐽 ∖ {𝑥}))) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))) | 
| 57 | 40, 56 | eqtrid 2788 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))) | 
| 58 | 57 | unieqd 4919 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ∪
((𝑆 ∘ 𝐹) “ (𝐽 ∖ {𝑥})) = ∪ (𝑆 “ (𝐼 ∖ {(𝐹‘𝑥)}))) | 
| 59 | 58 | fveq2d 6909 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
∘ 𝐹) “ (𝐽 ∖ {𝑥}))) = ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {(𝐹‘𝑥)})))) | 
| 60 | 39, 59 | ineq12d 4220 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {(𝐹‘𝑥)}))))) | 
| 61 | 4 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → 𝐺dom DProd 𝑆) | 
| 62 | 10 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → dom 𝑆 = 𝐼) | 
| 63 | 16 | ffvelcdmda 7103 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹‘𝑥) ∈ 𝐼) | 
| 64 | 61, 62, 63, 2, 3 | dprddisj 20030 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → ((𝑆‘(𝐹‘𝑥)) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {(𝐹‘𝑥)})))) = {(0g‘𝐺)}) | 
| 65 | 60, 64 | eqtrd 2776 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0g‘𝐺)}) | 
| 66 |  | eqimss 4041 | . . . 4
⊢ ((((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
∘ 𝐹) “ (𝐽 ∖ {𝑥})))) = {(0g‘𝐺)} → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
∘ 𝐹) “ (𝐽 ∖ {𝑥})))) ⊆ {(0g‘𝐺)}) | 
| 67 | 65, 66 | syl 17 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (((𝑆 ∘ 𝐹)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
∘ 𝐹) “ (𝐽 ∖ {𝑥})))) ⊆ {(0g‘𝐺)}) | 
| 68 | 1, 2, 3, 6, 13, 18, 38, 67 | dmdprdd 20020 | . 2
⊢ (𝜑 → 𝐺dom DProd (𝑆 ∘ 𝐹)) | 
| 69 |  | rnco2 6272 | . . . . . 6
⊢ ran
(𝑆 ∘ 𝐹) = (𝑆 “ ran 𝐹) | 
| 70 |  | forn 6822 | . . . . . . . . 9
⊢ (𝐹:𝐽–onto→𝐼 → ran 𝐹 = 𝐼) | 
| 71 | 7, 46, 70 | 3syl 18 | . . . . . . . 8
⊢ (𝜑 → ran 𝐹 = 𝐼) | 
| 72 | 71 | imaeq2d 6077 | . . . . . . 7
⊢ (𝜑 → (𝑆 “ ran 𝐹) = (𝑆 “ 𝐼)) | 
| 73 |  | ffn 6735 | . . . . . . . 8
⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → 𝑆 Fn 𝐼) | 
| 74 |  | fnima 6697 | . . . . . . . 8
⊢ (𝑆 Fn 𝐼 → (𝑆 “ 𝐼) = ran 𝑆) | 
| 75 | 14, 73, 74 | 3syl 18 | . . . . . . 7
⊢ (𝜑 → (𝑆 “ 𝐼) = ran 𝑆) | 
| 76 | 72, 75 | eqtrd 2776 | . . . . . 6
⊢ (𝜑 → (𝑆 “ ran 𝐹) = ran 𝑆) | 
| 77 | 69, 76 | eqtrid 2788 | . . . . 5
⊢ (𝜑 → ran (𝑆 ∘ 𝐹) = ran 𝑆) | 
| 78 | 77 | unieqd 4919 | . . . 4
⊢ (𝜑 → ∪ ran (𝑆 ∘ 𝐹) = ∪ ran 𝑆) | 
| 79 | 78 | fveq2d 6909 | . . 3
⊢ (𝜑 →
((mrCls‘(SubGrp‘𝐺))‘∪ ran
(𝑆 ∘ 𝐹)) =
((mrCls‘(SubGrp‘𝐺))‘∪ ran
𝑆)) | 
| 80 | 3 | dprdspan 20048 | . . . 4
⊢ (𝐺dom DProd (𝑆 ∘ 𝐹) → (𝐺 DProd (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹))) | 
| 81 | 68, 80 | syl 17 | . . 3
⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ∘ 𝐹))) | 
| 82 | 3 | dprdspan 20048 | . . . 4
⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆)) | 
| 83 | 4, 82 | syl 17 | . . 3
⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆)) | 
| 84 | 79, 81, 83 | 3eqtr4d 2786 | . 2
⊢ (𝜑 → (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆)) | 
| 85 | 68, 84 | jca 511 | 1
⊢ (𝜑 → (𝐺dom DProd (𝑆 ∘ 𝐹) ∧ (𝐺 DProd (𝑆 ∘ 𝐹)) = (𝐺 DProd 𝑆))) |