Step | Hyp | Ref
| Expression |
1 | | dprdsplit.u |
. . . . . 6
⊢ (𝜑 → 𝐼 = (𝐶 ∪ 𝐷)) |
2 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐼 = (𝐶 ∪ 𝐷)) |
3 | 2 | eleq2d 2673 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 ↔ 𝑌 ∈ (𝐶 ∪ 𝐷))) |
4 | | elun 3715 |
. . . 4
⊢ (𝑌 ∈ (𝐶 ∪ 𝐷) ↔ (𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷)) |
5 | 3, 4 | syl6bb 275 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 ↔ (𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷))) |
6 | | dmdprdsplit2.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐶)) |
7 | 6 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐶)) |
8 | | dprdsplit.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
9 | | ssun1 3738 |
. . . . . . . . . . 11
⊢ 𝐶 ⊆ (𝐶 ∪ 𝐷) |
10 | 9, 1 | syl5sseqr 3617 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ⊆ 𝐼) |
11 | 8, 10 | fssresd 5984 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺)) |
12 | | fdm 5964 |
. . . . . . . . 9
⊢ ((𝑆 ↾ 𝐶):𝐶⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐶) = 𝐶) |
13 | 11, 12 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → dom (𝑆 ↾ 𝐶) = 𝐶) |
14 | 13 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐶) = 𝐶) |
15 | | simplr 788 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐶) |
16 | | simprl 790 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝐶) |
17 | | simprr 792 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ≠ 𝑌) |
18 | | dmdprdsplit.z |
. . . . . . 7
⊢ 𝑍 = (Cntz‘𝐺) |
19 | 7, 14, 15, 16, 17, 18 | dprdcntz 18230 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝑍‘((𝑆 ↾ 𝐶)‘𝑌))) |
20 | | fvres 6117 |
. . . . . . 7
⊢ (𝑋 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋)) |
21 | 20 | ad2antlr 759 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋)) |
22 | | fvres 6117 |
. . . . . . . 8
⊢ (𝑌 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑌) = (𝑆‘𝑌)) |
23 | 22 | ad2antrl 760 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑌) = (𝑆‘𝑌)) |
24 | 23 | fveq2d 6107 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → (𝑍‘((𝑆 ↾ 𝐶)‘𝑌)) = (𝑍‘(𝑆‘𝑌))) |
25 | 19, 21, 24 | 3sstr3d 3610 |
. . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐶 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))) |
26 | 25 | exp32 629 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐶 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))) |
27 | 20 | ad2antlr 759 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋)) |
28 | 6 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐶)) |
29 | 13 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐶) = 𝐶) |
30 | | simplr 788 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝐶) |
31 | 28, 29, 30 | dprdub 18247 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) |
32 | 27, 31 | eqsstr3d 3603 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) |
33 | | dmdprdsplit2.3 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) |
34 | 33 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) |
35 | | eqid 2610 |
. . . . . . . . 9
⊢
(Base‘𝐺) =
(Base‘𝐺) |
36 | 35 | dprdssv 18238 |
. . . . . . . 8
⊢ (𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (Base‘𝐺) |
37 | | fvres 6117 |
. . . . . . . . . 10
⊢ (𝑌 ∈ 𝐷 → ((𝑆 ↾ 𝐷)‘𝑌) = (𝑆‘𝑌)) |
38 | 37 | ad2antrl 760 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐷)‘𝑌) = (𝑆‘𝑌)) |
39 | | dmdprdsplit2.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ 𝐷)) |
40 | 39 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝐺dom DProd (𝑆 ↾ 𝐷)) |
41 | | ssun2 3739 |
. . . . . . . . . . . . . 14
⊢ 𝐷 ⊆ (𝐶 ∪ 𝐷) |
42 | 41, 1 | syl5sseqr 3617 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐷 ⊆ 𝐼) |
43 | 8, 42 | fssresd 5984 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺)) |
44 | | fdm 5964 |
. . . . . . . . . . . 12
⊢ ((𝑆 ↾ 𝐷):𝐷⟶(SubGrp‘𝐺) → dom (𝑆 ↾ 𝐷) = 𝐷) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → dom (𝑆 ↾ 𝐷) = 𝐷) |
46 | 45 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → dom (𝑆 ↾ 𝐷) = 𝐷) |
47 | | simprl 790 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝐷) |
48 | 40, 46, 47 | dprdub 18247 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑆 ↾ 𝐷)‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) |
49 | 38, 48 | eqsstr3d 3603 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) |
50 | 35, 18 | cntz2ss 17588 |
. . . . . . . 8
⊢ (((𝐺 DProd (𝑆 ↾ 𝐷)) ⊆ (Base‘𝐺) ∧ (𝑆‘𝑌) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) → (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝑍‘(𝑆‘𝑌))) |
51 | 36, 49, 50 | sylancr 694 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ (𝑍‘(𝑆‘𝑌))) |
52 | 34, 51 | sstrd 3578 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝑆‘𝑌))) |
53 | 32, 52 | sstrd 3578 |
. . . . 5
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ (𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))) |
54 | 53 | exp32 629 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐷 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))) |
55 | 26, 54 | jaod 394 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐶 ∨ 𝑌 ∈ 𝐷) → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))) |
56 | 5, 55 | sylbid 229 |
. 2
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌))))) |
57 | | dprdgrp 18227 |
. . . . . . . 8
⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → 𝐺 ∈ Grp) |
58 | 6, 57 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Grp) |
59 | 58 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐺 ∈ Grp) |
60 | 35 | subgacs 17452 |
. . . . . 6
⊢ (𝐺 ∈ Grp →
(SubGrp‘𝐺) ∈
(ACS‘(Base‘𝐺))) |
61 | | acsmre 16136 |
. . . . . 6
⊢
((SubGrp‘𝐺)
∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
62 | 59, 60, 61 | 3syl 18 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
63 | | difundir 3839 |
. . . . . . . . . . 11
⊢ ((𝐶 ∪ 𝐷) ∖ {𝑋}) = ((𝐶 ∖ {𝑋}) ∪ (𝐷 ∖ {𝑋})) |
64 | 2 | difeq1d 3689 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐼 ∖ {𝑋}) = ((𝐶 ∪ 𝐷) ∖ {𝑋})) |
65 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐶) |
66 | 65 | snssd 4281 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → {𝑋} ⊆ 𝐶) |
67 | | sslin 3801 |
. . . . . . . . . . . . . . 15
⊢ ({𝑋} ⊆ 𝐶 → (𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶)) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶)) |
69 | | incom 3767 |
. . . . . . . . . . . . . . 15
⊢ (𝐶 ∩ 𝐷) = (𝐷 ∩ 𝐶) |
70 | | dprdsplit.i |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅) |
71 | 70 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐶 ∩ 𝐷) = ∅) |
72 | 69, 71 | syl5eqr 2658 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ 𝐶) = ∅) |
73 | | sseq0 3927 |
. . . . . . . . . . . . . 14
⊢ (((𝐷 ∩ {𝑋}) ⊆ (𝐷 ∩ 𝐶) ∧ (𝐷 ∩ 𝐶) = ∅) → (𝐷 ∩ {𝑋}) = ∅) |
74 | 68, 72, 73 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐷 ∩ {𝑋}) = ∅) |
75 | | disj3 3973 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ 𝐷 = (𝐷 ∖ {𝑋})) |
76 | 74, 75 | sylib 207 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐷 = (𝐷 ∖ {𝑋})) |
77 | 76 | uneq2d 3729 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐶 ∖ {𝑋}) ∪ 𝐷) = ((𝐶 ∖ {𝑋}) ∪ (𝐷 ∖ {𝑋}))) |
78 | 63, 64, 77 | 3eqtr4a 2670 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐼 ∖ {𝑋}) = ((𝐶 ∖ {𝑋}) ∪ 𝐷)) |
79 | 78 | imaeq2d 5385 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ (𝐼 ∖ {𝑋})) = (𝑆 “ ((𝐶 ∖ {𝑋}) ∪ 𝐷))) |
80 | | imaundi 5464 |
. . . . . . . . 9
⊢ (𝑆 “ ((𝐶 ∖ {𝑋}) ∪ 𝐷)) = ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)) |
81 | 79, 80 | syl6eq 2660 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ (𝐼 ∖ {𝑋})) = ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷))) |
82 | 81 | unieqd 4382 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) = ∪ ((𝑆 “ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷))) |
83 | | uniun 4392 |
. . . . . . 7
⊢ ∪ ((𝑆
“ (𝐶 ∖ {𝑋})) ∪ (𝑆 “ 𝐷)) = (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪
(𝑆 “ 𝐷)) |
84 | 82, 83 | syl6eq 2660 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) = (∪ (𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪
(𝑆 “ 𝐷))) |
85 | | dmdprdsplit2lem.k |
. . . . . . . . 9
⊢ 𝐾 =
(mrCls‘(SubGrp‘𝐺)) |
86 | | difss 3699 |
. . . . . . . . . . 11
⊢ (𝐶 ∖ {𝑋}) ⊆ 𝐶 |
87 | | imass2 5420 |
. . . . . . . . . . 11
⊢ ((𝐶 ∖ {𝑋}) ⊆ 𝐶 → (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑆 “ 𝐶)) |
88 | | uniss 4394 |
. . . . . . . . . . 11
⊢ ((𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑆 “ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ ∪
(𝑆 “ 𝐶)) |
89 | 86, 87, 88 | mp2b 10 |
. . . . . . . . . 10
⊢ ∪ (𝑆
“ (𝐶 ∖ {𝑋})) ⊆ ∪ (𝑆
“ 𝐶) |
90 | | imassrn 5396 |
. . . . . . . . . . . 12
⊢ (𝑆 “ 𝐶) ⊆ ran 𝑆 |
91 | | frn 5966 |
. . . . . . . . . . . . . . 15
⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺)) |
92 | 8, 91 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺)) |
93 | 92 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ran 𝑆 ⊆ (SubGrp‘𝐺)) |
94 | | mresspw 16075 |
. . . . . . . . . . . . . 14
⊢
((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)) |
95 | 62, 94 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)) |
96 | 93, 95 | sstrd 3578 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺)) |
97 | 90, 96 | syl5ss 3579 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ 𝐶) ⊆ 𝒫 (Base‘𝐺)) |
98 | | sspwuni 4547 |
. . . . . . . . . . 11
⊢ ((𝑆 “ 𝐶) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆
“ 𝐶) ⊆
(Base‘𝐺)) |
99 | 97, 98 | sylib 207 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐶) ⊆ (Base‘𝐺)) |
100 | 89, 99 | syl5ss 3579 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (Base‘𝐺)) |
101 | 62, 85, 100 | mrcssidd 16108 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) |
102 | | imassrn 5396 |
. . . . . . . . . . . 12
⊢ (𝑆 “ 𝐷) ⊆ ran 𝑆 |
103 | 102, 96 | syl5ss 3579 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆 “ 𝐷) ⊆ 𝒫 (Base‘𝐺)) |
104 | | sspwuni 4547 |
. . . . . . . . . . 11
⊢ ((𝑆 “ 𝐷) ⊆ 𝒫 (Base‘𝐺) ↔ ∪ (𝑆
“ 𝐷) ⊆
(Base‘𝐺)) |
105 | 103, 104 | sylib 207 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (Base‘𝐺)) |
106 | 62, 85, 105 | mrcssidd 16108 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (𝐾‘∪ (𝑆 “ 𝐷))) |
107 | 85 | dprdspan 18249 |
. . . . . . . . . . . 12
⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ ran
(𝑆 ↾ 𝐷))) |
108 | 39, 107 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ ran
(𝑆 ↾ 𝐷))) |
109 | | df-ima 5051 |
. . . . . . . . . . . . 13
⊢ (𝑆 “ 𝐷) = ran (𝑆 ↾ 𝐷) |
110 | 109 | unieqi 4381 |
. . . . . . . . . . . 12
⊢ ∪ (𝑆
“ 𝐷) = ∪ ran (𝑆 ↾ 𝐷) |
111 | 110 | fveq2i 6106 |
. . . . . . . . . . 11
⊢ (𝐾‘∪ (𝑆
“ 𝐷)) = (𝐾‘∪ ran (𝑆 ↾ 𝐷)) |
112 | 108, 111 | syl6eqr 2662 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ (𝑆 “ 𝐷))) |
113 | 112 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐷)) = (𝐾‘∪ (𝑆 “ 𝐷))) |
114 | 106, 113 | sseqtr4d 3605 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ 𝐷) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) |
115 | | unss12 3747 |
. . . . . . . 8
⊢ ((∪ (𝑆
“ (𝐶 ∖ {𝑋})) ⊆ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∧ ∪
(𝑆 “ 𝐷) ⊆ (𝐺 DProd (𝑆 ↾ 𝐷))) → (∪
(𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪
(𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷)))) |
116 | 101, 114,
115 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (∪
(𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪
(𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷)))) |
117 | 85 | mrccl 16094 |
. . . . . . . . 9
⊢
(((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (Base‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) |
118 | 62, 100, 117 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) |
119 | | dprdsubg 18246 |
. . . . . . . . . 10
⊢ (𝐺dom DProd (𝑆 ↾ 𝐷) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) |
120 | 39, 119 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) |
121 | 120 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) |
122 | | eqid 2610 |
. . . . . . . . 9
⊢
(LSSum‘𝐺) =
(LSSum‘𝐺) |
123 | 122 | lsmunss 17896 |
. . . . . . . 8
⊢ (((𝐾‘∪ (𝑆
“ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺)) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) |
124 | 118, 121,
123 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∪ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) |
125 | 116, 124 | sstrd 3578 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (∪
(𝑆 “ (𝐶 ∖ {𝑋})) ∪ ∪
(𝑆 “ 𝐷)) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) |
126 | 84, 125 | eqsstrd 3602 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐼 ∖ {𝑋})) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) |
127 | 89 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ ∪
(𝑆 “ 𝐶)) |
128 | 62, 85, 127, 99 | mrcssd 16107 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐾‘∪ (𝑆 “ 𝐶))) |
129 | 85 | dprdspan 18249 |
. . . . . . . . . . 11
⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ ran
(𝑆 ↾ 𝐶))) |
130 | 6, 129 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ ran
(𝑆 ↾ 𝐶))) |
131 | | df-ima 5051 |
. . . . . . . . . . . 12
⊢ (𝑆 “ 𝐶) = ran (𝑆 ↾ 𝐶) |
132 | 131 | unieqi 4381 |
. . . . . . . . . . 11
⊢ ∪ (𝑆
“ 𝐶) = ∪ ran (𝑆 ↾ 𝐶) |
133 | 132 | fveq2i 6106 |
. . . . . . . . . 10
⊢ (𝐾‘∪ (𝑆
“ 𝐶)) = (𝐾‘∪ ran (𝑆 ↾ 𝐶)) |
134 | 130, 133 | syl6eqr 2662 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ (𝑆 “ 𝐶))) |
135 | 134 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) = (𝐾‘∪ (𝑆 “ 𝐶))) |
136 | 128, 135 | sseqtr4d 3605 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) |
137 | 33 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) |
138 | 136, 137 | sstrd 3578 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) |
139 | 122, 18 | lsmsubg 17892 |
. . . . . 6
⊢ (((𝐾‘∪ (𝑆
“ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝐺 DProd (𝑆 ↾ 𝐷)))) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺)) |
140 | 118, 121,
138, 139 | syl3anc 1318 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺)) |
141 | 85 | mrcsscl 16103 |
. . . . 5
⊢
(((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐼 ∖ {𝑋})) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∧ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) ∈ (SubGrp‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) |
142 | 62, 126, 140, 141 | syl3anc 1318 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) |
143 | | sslin 3801 |
. . . 4
⊢ ((𝐾‘∪ (𝑆
“ (𝐼 ∖ {𝑋}))) ⊆ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))) |
144 | 142, 143 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷))))) |
145 | 10 | sselda 3568 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝐼) |
146 | 8 | ffvelrnda 6267 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐼) → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
147 | 145, 146 | syldan 486 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
148 | | dmdprdsplit.0 |
. . . 4
⊢ 0 =
(0g‘𝐺) |
149 | 20 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋)) |
150 | 6 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝐺dom DProd (𝑆 ↾ 𝐶)) |
151 | 13 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → dom (𝑆 ↾ 𝐶) = 𝐶) |
152 | 150, 151,
65 | dprdub 18247 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶)‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) |
153 | 149, 152 | eqsstr3d 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) |
154 | | dprdsubg 18246 |
. . . . . . . . . . 11
⊢ (𝐺dom DProd (𝑆 ↾ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺)) |
155 | 6, 154 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺)) |
156 | 155 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺)) |
157 | 122 | lsmlub 17901 |
. . . . . . . . 9
⊢ (((𝑆‘𝑋) ∈ (SubGrp‘𝐺) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑆 ↾ 𝐶)) ∈ (SubGrp‘𝐺)) → (((𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) ↔ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))) |
158 | 147, 118,
156, 157 | syl3anc 1318 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) ↔ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)))) |
159 | 153, 136,
158 | mpbi2and 958 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶))) |
160 | | ssrin 3800 |
. . . . . . 7
⊢ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ⊆ (𝐺 DProd (𝑆 ↾ 𝐶)) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷)))) |
161 | 159, 160 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷)))) |
162 | | dmdprdsplit2.4 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) |
163 | 162 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝐺 DProd (𝑆 ↾ 𝐶)) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) |
164 | 161, 163 | sseqtrd 3604 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) ⊆ { 0 }) |
165 | 122 | lsmub1 17894 |
. . . . . . . . 9
⊢ (((𝑆‘𝑋) ∈ (SubGrp‘𝐺) ∧ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) → (𝑆‘𝑋) ⊆ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))) |
166 | 147, 118,
165 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))) |
167 | 148 | subg0cl 17425 |
. . . . . . . . 9
⊢ ((𝑆‘𝑋) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑋)) |
168 | 147, 167 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (𝑆‘𝑋)) |
169 | 166, 168 | sseldd 3569 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ ((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))) |
170 | 148 | subg0cl 17425 |
. . . . . . . 8
⊢ ((𝐺 DProd (𝑆 ↾ 𝐷)) ∈ (SubGrp‘𝐺) → 0 ∈ (𝐺 DProd (𝑆 ↾ 𝐷))) |
171 | 121, 170 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (𝐺 DProd (𝑆 ↾ 𝐷))) |
172 | 169, 171 | elind 3760 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 0 ∈ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷)))) |
173 | 172 | snssd 4281 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → { 0 } ⊆ (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷)))) |
174 | 164, 173 | eqssd 3585 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆‘𝑋)(LSSum‘𝐺)(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) ∩ (𝐺 DProd (𝑆 ↾ 𝐷))) = { 0 }) |
175 | | resima2 5352 |
. . . . . . . . 9
⊢ ((𝐶 ∖ {𝑋}) ⊆ 𝐶 → ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = (𝑆 “ (𝐶 ∖ {𝑋}))) |
176 | 86, 175 | mp1i 13 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = (𝑆 “ (𝐶 ∖ {𝑋}))) |
177 | 176 | unieqd 4382 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪
((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})) = ∪ (𝑆 “ (𝐶 ∖ {𝑋}))) |
178 | 177 | fveq2d 6107 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋}))) = (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) |
179 | 149, 178 | ineq12d 3777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆 ↾ 𝐶)‘𝑋) ∩ (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})))) = ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))) |
180 | 150, 151,
65, 148, 85 | dprddisj 18231 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (((𝑆 ↾ 𝐶)‘𝑋) ∩ (𝐾‘∪ ((𝑆 ↾ 𝐶) “ (𝐶 ∖ {𝑋})))) = { 0 }) |
181 | 179, 180 | eqtr3d 2646 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))) = { 0 }) |
182 | 8 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
183 | | ffun 5961 |
. . . . . . . 8
⊢ (𝑆:𝐼⟶(SubGrp‘𝐺) → Fun 𝑆) |
184 | | funiunfv 6410 |
. . . . . . . 8
⊢ (Fun
𝑆 → ∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) = ∪ (𝑆 “ (𝐶 ∖ {𝑋}))) |
185 | 182, 183,
184 | 3syl 18 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪
𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) = ∪ (𝑆 “ (𝐶 ∖ {𝑋}))) |
186 | 6 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝐺dom DProd (𝑆 ↾ 𝐶)) |
187 | 13 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → dom (𝑆 ↾ 𝐶) = 𝐶) |
188 | | eldifi 3694 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (𝐶 ∖ {𝑋}) → 𝑦 ∈ 𝐶) |
189 | 188 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑦 ∈ 𝐶) |
190 | | simplr 788 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑋 ∈ 𝐶) |
191 | | eldifsni 4261 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (𝐶 ∖ {𝑋}) → 𝑦 ≠ 𝑋) |
192 | 191 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → 𝑦 ≠ 𝑋) |
193 | 186, 187,
189, 190, 192, 18 | dprdcntz 18230 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑦) ⊆ (𝑍‘((𝑆 ↾ 𝐶)‘𝑋))) |
194 | | fvres 6117 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝐶 → ((𝑆 ↾ 𝐶)‘𝑦) = (𝑆‘𝑦)) |
195 | 189, 194 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑦) = (𝑆‘𝑦)) |
196 | 20 | ad2antlr 759 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → ((𝑆 ↾ 𝐶)‘𝑋) = (𝑆‘𝑋)) |
197 | 196 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → (𝑍‘((𝑆 ↾ 𝐶)‘𝑋)) = (𝑍‘(𝑆‘𝑋))) |
198 | 193, 195,
197 | 3sstr3d 3610 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ∈ 𝐶) ∧ 𝑦 ∈ (𝐶 ∖ {𝑋})) → (𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋))) |
199 | 198 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∀𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋))) |
200 | | iunss 4497 |
. . . . . . . 8
⊢ (∪ 𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋))) |
201 | 199, 200 | sylibr 223 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪
𝑦 ∈ (𝐶 ∖ {𝑋})(𝑆‘𝑦) ⊆ (𝑍‘(𝑆‘𝑋))) |
202 | 185, 201 | eqsstr3d 3603 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑍‘(𝑆‘𝑋))) |
203 | 35 | subgss 17418 |
. . . . . . . 8
⊢ ((𝑆‘𝑋) ∈ (SubGrp‘𝐺) → (𝑆‘𝑋) ⊆ (Base‘𝐺)) |
204 | 147, 203 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (Base‘𝐺)) |
205 | 35, 18 | cntzsubg 17592 |
. . . . . . 7
⊢ ((𝐺 ∈ Grp ∧ (𝑆‘𝑋) ⊆ (Base‘𝐺)) → (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺)) |
206 | 59, 204, 205 | syl2anc 691 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺)) |
207 | 85 | mrcsscl 16103 |
. . . . . 6
⊢
(((SubGrp‘𝐺)
∈ (Moore‘(Base‘𝐺)) ∧ ∪ (𝑆 “ (𝐶 ∖ {𝑋})) ⊆ (𝑍‘(𝑆‘𝑋)) ∧ (𝑍‘(𝑆‘𝑋)) ∈ (SubGrp‘𝐺)) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝑆‘𝑋))) |
208 | 62, 202, 206, 207 | syl3anc 1318 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))) ⊆ (𝑍‘(𝑆‘𝑋))) |
209 | 18, 118, 147, 208 | cntzrecd 17914 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → (𝑆‘𝑋) ⊆ (𝑍‘(𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋}))))) |
210 | 122, 147,
118, 121, 148, 174, 181, 18, 209 | lsmdisj3 17919 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ ((𝐾‘∪ (𝑆 “ (𝐶 ∖ {𝑋})))(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ 𝐷)))) = { 0 }) |
211 | 144, 210 | sseqtrd 3604 |
. 2
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 }) |
212 | 56, 211 | jca 553 |
1
⊢ ((𝜑 ∧ 𝑋 ∈ 𝐶) → ((𝑌 ∈ 𝐼 → (𝑋 ≠ 𝑌 → (𝑆‘𝑋) ⊆ (𝑍‘(𝑆‘𝑌)))) ∧ ((𝑆‘𝑋) ∩ (𝐾‘∪ (𝑆 “ (𝐼 ∖ {𝑋})))) ⊆ { 0 })) |