Step | Hyp | Ref
| Expression |
1 | | dprdres.1 |
. . . 4
⊢ (𝜑 → 𝐺dom DProd 𝑆) |
2 | | dprdgrp 18227 |
. . . 4
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
3 | 1, 2 | syl 17 |
. . 3
⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | | dprdres.2 |
. . . . 5
⊢ (𝜑 → dom 𝑆 = 𝐼) |
5 | 1, 4 | dprdf2 18229 |
. . . 4
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
6 | | dprdres.3 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ 𝐼) |
7 | 5, 6 | fssresd 5984 |
. . 3
⊢ (𝜑 → (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺)) |
8 | 1 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆) |
9 | 4 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼) |
10 | 6 | ad2antrr 758 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴 ⊆ 𝐼) |
11 | | simplr 788 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐴) |
12 | 10, 11 | sseldd 3569 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ∈ 𝐼) |
13 | | eldifi 3694 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ∈ 𝐴) |
14 | 13 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐴) |
15 | 10, 14 | sseldd 3569 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ∈ 𝐼) |
16 | | eldifsni 4261 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦 ≠ 𝑥) |
17 | 16 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦 ≠ 𝑥) |
18 | 17 | necomd 2837 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥 ≠ 𝑦) |
19 | | eqid 2610 |
. . . . . . . 8
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
20 | 8, 9, 12, 15, 18, 19 | dprdcntz 18230 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦))) |
21 | | fvres 6117 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥)) |
22 | 11, 21 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥)) |
23 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝐴 → ((𝑆 ↾ 𝐴)‘𝑦) = (𝑆‘𝑦)) |
24 | 14, 23 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑦) = (𝑆‘𝑦)) |
25 | 24 | fveq2d 6107 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆‘𝑦))) |
26 | 20, 22, 25 | 3sstr4d 3611 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦))) |
27 | 26 | ralrimiva 2949 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦))) |
28 | 21 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴)‘𝑥) = (𝑆‘𝑥)) |
29 | 28 | ineq1d 3775 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥}))))) |
30 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢
(Base‘𝐺) =
(Base‘𝐺) |
31 | 30 | subgacs 17452 |
. . . . . . . . . . . 12
⊢ (𝐺 ∈ Grp →
(SubGrp‘𝐺) ∈
(ACS‘(Base‘𝐺))) |
32 | | acsmre 16136 |
. . . . . . . . . . . 12
⊢
((SubGrp‘𝐺)
∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
33 | 3, 31, 32 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (SubGrp‘𝐺) ∈
(Moore‘(Base‘𝐺))) |
34 | 33 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
35 | | eqid 2610 |
. . . . . . . . . 10
⊢
(mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺)) |
36 | | resss 5342 |
. . . . . . . . . . . . 13
⊢ (𝑆 ↾ 𝐴) ⊆ 𝑆 |
37 | | imass1 5419 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))) |
38 | 36, 37 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})) |
39 | 6 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ 𝐼) |
40 | | ssdif 3707 |
. . . . . . . . . . . . 13
⊢ (𝐴 ⊆ 𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥})) |
41 | | imass2 5420 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥}))) |
42 | 39, 40, 41 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥}))) |
43 | 38, 42 | syl5ss 3579 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥}))) |
44 | 43 | unissd 4398 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ ∪
(𝑆 “ (𝐼 ∖ {𝑥}))) |
45 | | imassrn 5396 |
. . . . . . . . . . . 12
⊢ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆 |
46 | | frn 5966 |
. . . . . . . . . . . . . . 15
⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺)) |
47 | 5, 46 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺)) |
48 | 30 | subgss 17418 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺)) |
49 | | selpw 4115 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ 𝒫
(Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺)) |
50 | 48, 49 | sylibr 223 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺)) |
51 | 50 | ssriv 3572 |
. . . . . . . . . . . . . 14
⊢
(SubGrp‘𝐺)
⊆ 𝒫 (Base‘𝐺) |
52 | 47, 51 | syl6ss 3580 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺)) |
53 | 52 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺)) |
54 | 45, 53 | syl5ss 3579 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺)) |
55 | | sspwuni 4547 |
. . . . . . . . . . 11
⊢ ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆
“ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) |
56 | 54, 55 | sylib 207 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺)) |
57 | 34, 35, 44, 56 | mrcssd 16107 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥})))) |
58 | | sslin 3801 |
. . . . . . . . 9
⊢
(((mrCls‘(SubGrp‘𝐺))‘∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ⊆
((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (𝐼 ∖ {𝑥}))) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))))) |
59 | 57, 58 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥}))))) |
60 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺dom DProd 𝑆) |
61 | 4 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom 𝑆 = 𝐼) |
62 | 6 | sselda 3568 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐼) |
63 | | eqid 2610 |
. . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) |
64 | 60, 61, 62, 63, 35 | dprddisj 18231 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆
“ (𝐼 ∖ {𝑥})))) =
{(0g‘𝐺)}) |
65 | 59, 64 | sseqtrd 3604 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g‘𝐺)}) |
66 | 5 | ffvelrnda 6267 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
67 | 62, 66 | syldan 486 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
68 | 63 | subg0cl 17425 |
. . . . . . . . . 10
⊢ ((𝑆‘𝑥) ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ (𝑆‘𝑥)) |
69 | 67, 68 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ (𝑆‘𝑥)) |
70 | 44, 56 | sstrd 3578 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) |
71 | 35 | mrccl 16094 |
. . . . . . . . . . 11
⊢
(((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) ∧ ∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
72 | 34, 70, 71 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
73 | 63 | subg0cl 17425 |
. . . . . . . . . 10
⊢
(((mrCls‘(SubGrp‘𝐺))‘∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈
((mrCls‘(SubGrp‘𝐺))‘∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) |
74 | 72, 73 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈
((mrCls‘(SubGrp‘𝐺))‘∪
((𝑆 ↾ 𝐴) “ (𝐴 ∖ {𝑥})))) |
75 | 69, 74 | elind 3760 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0g‘𝐺) ∈ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥}))))) |
76 | 75 | snssd 4281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {(0g‘𝐺)} ⊆ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥}))))) |
77 | 65, 76 | eqssd 3585 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)}) |
78 | 29, 77 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)}) |
79 | 27, 78 | jca 553 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})) |
80 | 79 | ralrimiva 2949 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})) |
81 | 1, 4 | dprddomcld 18223 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ V) |
82 | 81, 6 | ssexd 4733 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ V) |
83 | | fdm 5964 |
. . . . 5
⊢ ((𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐴) = 𝐴) |
84 | 7, 83 | syl 17 |
. . . 4
⊢ (𝜑 → dom (𝑆 ↾ 𝐴) = 𝐴) |
85 | 19, 63, 35 | dmdprd 18220 |
. . . 4
⊢ ((𝐴 ∈ V ∧ dom (𝑆 ↾ 𝐴) = 𝐴) → (𝐺dom DProd (𝑆 ↾ 𝐴) ↔ (𝐺 ∈ Grp ∧ (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})))) |
86 | 82, 84, 85 | syl2anc 691 |
. . 3
⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ↔ (𝐺 ∈ Grp ∧ (𝑆 ↾ 𝐴):𝐴⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆 ↾ 𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆 ↾ 𝐴)‘𝑦)) ∧ (((𝑆 ↾ 𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ ((𝑆
↾ 𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g‘𝐺)})))) |
87 | 3, 7, 80, 86 | mpbir3and 1238 |
. 2
⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐴)) |
88 | | rnss 5275 |
. . . . . 6
⊢ ((𝑆 ↾ 𝐴) ⊆ 𝑆 → ran (𝑆 ↾ 𝐴) ⊆ ran 𝑆) |
89 | | uniss 4394 |
. . . . . 6
⊢ (ran
(𝑆 ↾ 𝐴) ⊆ ran 𝑆 → ∪ ran
(𝑆 ↾ 𝐴) ⊆ ∪ ran 𝑆) |
90 | 36, 88, 89 | mp2b 10 |
. . . . 5
⊢ ∪ ran (𝑆 ↾ 𝐴) ⊆ ∪ ran
𝑆 |
91 | 90 | a1i 11 |
. . . 4
⊢ (𝜑 → ∪ ran (𝑆 ↾ 𝐴) ⊆ ∪ ran
𝑆) |
92 | | sspwuni 4547 |
. . . . 5
⊢ (ran
𝑆 ⊆ 𝒫
(Base‘𝐺) ↔ ∪ ran 𝑆 ⊆ (Base‘𝐺)) |
93 | 52, 92 | sylib 207 |
. . . 4
⊢ (𝜑 → ∪ ran 𝑆 ⊆ (Base‘𝐺)) |
94 | 33, 35, 91, 93 | mrcssd 16107 |
. . 3
⊢ (𝜑 →
((mrCls‘(SubGrp‘𝐺))‘∪ ran
(𝑆 ↾ 𝐴)) ⊆
((mrCls‘(SubGrp‘𝐺))‘∪ ran
𝑆)) |
95 | 35 | dprdspan 18249 |
. . . 4
⊢ (𝐺dom DProd (𝑆 ↾ 𝐴) → (𝐺 DProd (𝑆 ↾ 𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴))) |
96 | 87, 95 | syl 17 |
. . 3
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran (𝑆 ↾ 𝐴))) |
97 | 35 | dprdspan 18249 |
. . . 4
⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆)) |
98 | 1, 97 | syl 17 |
. . 3
⊢ (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘∪ ran 𝑆)) |
99 | 94, 96, 98 | 3sstr4d 3611 |
. 2
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐴)) ⊆ (𝐺 DProd 𝑆)) |
100 | 87, 99 | jca 553 |
1
⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ 𝐴) ∧ (𝐺 DProd (𝑆 ↾ 𝐴)) ⊆ (𝐺 DProd 𝑆))) |