Step | Hyp | Ref
| Expression |
1 | | ssid 3624 |
. 2
⊢ 𝐴 ⊆ 𝐴 |
2 | | ablfac1eulem.2 |
. . 3
⊢ (𝜑 → 𝐴 ∈ Fin) |
3 | | sseq1 3626 |
. . . . . 6
⊢ (𝑦 = ∅ → (𝑦 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
4 | | difeq1 3721 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ∅ → (𝑦 ∖ {𝑃}) = (∅ ∖ {𝑃})) |
5 | | 0dif 3977 |
. . . . . . . . . . . . 13
⊢ (∅
∖ {𝑃}) =
∅ |
6 | 4, 5 | syl6eq 2672 |
. . . . . . . . . . . 12
⊢ (𝑦 = ∅ → (𝑦 ∖ {𝑃}) = ∅) |
7 | 6 | reseq2d 5396 |
. . . . . . . . . . 11
⊢ (𝑦 = ∅ → (𝑇 ↾ (𝑦 ∖ {𝑃})) = (𝑇 ↾ ∅)) |
8 | | res0 5400 |
. . . . . . . . . . 11
⊢ (𝑇 ↾ ∅) =
∅ |
9 | 7, 8 | syl6eq 2672 |
. . . . . . . . . 10
⊢ (𝑦 = ∅ → (𝑇 ↾ (𝑦 ∖ {𝑃})) = ∅) |
10 | 9 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑦 = ∅ → (𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))) = (𝐺 DProd ∅)) |
11 | 10 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑦 = ∅ →
(#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) = (#‘(𝐺 DProd ∅))) |
12 | 11 | breq2d 4665 |
. . . . . . 7
⊢ (𝑦 = ∅ → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ 𝑃 ∥ (#‘(𝐺 DProd ∅)))) |
13 | 12 | notbid 308 |
. . . . . 6
⊢ (𝑦 = ∅ → (¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ ¬ 𝑃 ∥ (#‘(𝐺 DProd ∅)))) |
14 | 3, 13 | imbi12d 334 |
. . . . 5
⊢ (𝑦 = ∅ → ((𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))))) ↔ (∅ ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd ∅))))) |
15 | 14 | imbi2d 330 |
. . . 4
⊢ (𝑦 = ∅ → ((𝜑 → (𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))))) ↔ (𝜑 → (∅ ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd ∅)))))) |
16 | | sseq1 3626 |
. . . . . 6
⊢ (𝑦 = 𝑧 → (𝑦 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴)) |
17 | | difeq1 3721 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑧 → (𝑦 ∖ {𝑃}) = (𝑧 ∖ {𝑃})) |
18 | 17 | reseq2d 5396 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑧 → (𝑇 ↾ (𝑦 ∖ {𝑃})) = (𝑇 ↾ (𝑧 ∖ {𝑃}))) |
19 | 18 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑦 = 𝑧 → (𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))) = (𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) |
20 | 19 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑦 = 𝑧 → (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) = (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) |
21 | 20 | breq2d 4665 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) |
22 | 21 | notbid 308 |
. . . . . 6
⊢ (𝑦 = 𝑧 → (¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) |
23 | 16, 22 | imbi12d 334 |
. . . . 5
⊢ (𝑦 = 𝑧 → ((𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))))) ↔ (𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))))) |
24 | 23 | imbi2d 330 |
. . . 4
⊢ (𝑦 = 𝑧 → ((𝜑 → (𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))))) ↔ (𝜑 → (𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))))) |
25 | | sseq1 3626 |
. . . . . 6
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (𝑦 ⊆ 𝐴 ↔ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) |
26 | | difeq1 3721 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (𝑦 ∖ {𝑃}) = ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
27 | 26 | reseq2d 5396 |
. . . . . . . . . 10
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (𝑇 ↾ (𝑦 ∖ {𝑃})) = (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))) |
28 | 27 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))) = (𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))) |
29 | 28 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) = (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))) |
30 | 29 | breq2d 4665 |
. . . . . . 7
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))) |
31 | 30 | notbid 308 |
. . . . . 6
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → (¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))) |
32 | 25, 31 | imbi12d 334 |
. . . . 5
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → ((𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))))) ↔ ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))))) |
33 | 32 | imbi2d 330 |
. . . 4
⊢ (𝑦 = (𝑧 ∪ {𝑞}) → ((𝜑 → (𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))))) ↔ (𝜑 → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))))) |
34 | | sseq1 3626 |
. . . . . 6
⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) |
35 | | difeq1 3721 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ∖ {𝑃}) = (𝐴 ∖ {𝑃})) |
36 | 35 | reseq2d 5396 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → (𝑇 ↾ (𝑦 ∖ {𝑃})) = (𝑇 ↾ (𝐴 ∖ {𝑃}))) |
37 | 36 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → (𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))) = (𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))) |
38 | 37 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑦 = 𝐴 → (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) = (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃}))))) |
39 | 38 | breq2d 4665 |
. . . . . . 7
⊢ (𝑦 = 𝐴 → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))) |
40 | 39 | notbid 308 |
. . . . . 6
⊢ (𝑦 = 𝐴 → (¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))) ↔ ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))) |
41 | 34, 40 | imbi12d 334 |
. . . . 5
⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃}))))) ↔ (𝐴 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃}))))))) |
42 | 41 | imbi2d 330 |
. . . 4
⊢ (𝑦 = 𝐴 → ((𝜑 → (𝑦 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑦 ∖ {𝑃})))))) ↔ (𝜑 → (𝐴 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))))) |
43 | | ablfac1eulem.1 |
. . . . . . 7
⊢ (𝜑 → 𝑃 ∈ ℙ) |
44 | | nprmdvds1 15418 |
. . . . . . 7
⊢ (𝑃 ∈ ℙ → ¬
𝑃 ∥
1) |
45 | 43, 44 | syl 17 |
. . . . . 6
⊢ (𝜑 → ¬ 𝑃 ∥ 1) |
46 | | ablfac1.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺 ∈ Abel) |
47 | | ablgrp 18198 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) |
48 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(0g‘𝐺) = (0g‘𝐺) |
49 | 48 | dprd0 18430 |
. . . . . . . . . . 11
⊢ (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) =
{(0g‘𝐺)})) |
50 | 46, 47, 49 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) =
{(0g‘𝐺)})) |
51 | 50 | simprd 479 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 DProd ∅) =
{(0g‘𝐺)}) |
52 | 51 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝜑 → (#‘(𝐺 DProd ∅)) =
(#‘{(0g‘𝐺)})) |
53 | | fvex 6201 |
. . . . . . . . 9
⊢
(0g‘𝐺) ∈ V |
54 | | hashsng 13159 |
. . . . . . . . 9
⊢
((0g‘𝐺) ∈ V →
(#‘{(0g‘𝐺)}) = 1) |
55 | 53, 54 | ax-mp 5 |
. . . . . . . 8
⊢
(#‘{(0g‘𝐺)}) = 1 |
56 | 52, 55 | syl6eq 2672 |
. . . . . . 7
⊢ (𝜑 → (#‘(𝐺 DProd ∅)) =
1) |
57 | 56 | breq2d 4665 |
. . . . . 6
⊢ (𝜑 → (𝑃 ∥ (#‘(𝐺 DProd ∅)) ↔ 𝑃 ∥ 1)) |
58 | 45, 57 | mtbird 315 |
. . . . 5
⊢ (𝜑 → ¬ 𝑃 ∥ (#‘(𝐺 DProd ∅))) |
59 | 58 | a1d 25 |
. . . 4
⊢ (𝜑 → (∅ ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd ∅)))) |
60 | | ssun1 3776 |
. . . . . . . . . 10
⊢ 𝑧 ⊆ (𝑧 ∪ {𝑞}) |
61 | | sstr 3611 |
. . . . . . . . . 10
⊢ ((𝑧 ⊆ (𝑧 ∪ {𝑞}) ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴) → 𝑧 ⊆ 𝐴) |
62 | 60, 61 | mpan 706 |
. . . . . . . . 9
⊢ ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → 𝑧 ⊆ 𝐴) |
63 | 62 | imim1i 63 |
. . . . . . . 8
⊢ ((𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) |
64 | | ablfac1eu.1 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐺dom DProd 𝑇 ∧ (𝐺 DProd 𝑇) = 𝐵)) |
65 | 64 | simpld 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐺dom DProd 𝑇) |
66 | | ablfac1eu.2 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → dom 𝑇 = 𝐴) |
67 | 65, 66 | dprdf2 18406 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑇:𝐴⟶(SubGrp‘𝐺)) |
68 | 67 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝑇:𝐴⟶(SubGrp‘𝐺)) |
69 | | simprr 796 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑧 ∪ {𝑞}) ⊆ 𝐴) |
70 | 69 | ssdifssd 3748 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑧 ∪ {𝑞}) ∖ {𝑃}) ⊆ 𝐴) |
71 | 68, 70 | fssresd 6071 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})):((𝑧 ∪ {𝑞}) ∖ {𝑃})⟶(SubGrp‘𝐺)) |
72 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ¬ 𝑞 ∈ 𝑧) |
73 | | disjsn 4246 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑧 ∩ {𝑞}) = ∅ ↔ ¬ 𝑞 ∈ 𝑧) |
74 | 72, 73 | sylibr 224 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑧 ∩ {𝑞}) = ∅) |
75 | 74 | difeq1d 3727 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑧 ∩ {𝑞}) ∖ {𝑃}) = (∅ ∖ {𝑃})) |
76 | | difindir 3882 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 ∩ {𝑞}) ∖ {𝑃}) = ((𝑧 ∖ {𝑃}) ∩ ({𝑞} ∖ {𝑃})) |
77 | 75, 76, 5 | 3eqtr3g 2679 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑧 ∖ {𝑃}) ∩ ({𝑞} ∖ {𝑃})) = ∅) |
78 | | difundir 3880 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 ∪ {𝑞}) ∖ {𝑃}) = ((𝑧 ∖ {𝑃}) ∪ ({𝑞} ∖ {𝑃})) |
79 | 78 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑧 ∪ {𝑞}) ∖ {𝑃}) = ((𝑧 ∖ {𝑃}) ∪ ({𝑞} ∖ {𝑃}))) |
80 | | eqid 2622 |
. . . . . . . . . . . . . . . . 17
⊢
(LSSum‘𝐺) =
(LSSum‘𝐺) |
81 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐺dom DProd 𝑇) |
82 | 66 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → dom 𝑇 = 𝐴) |
83 | 81, 82, 70 | dprdres 18427 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺dom DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ∧ (𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))) ⊆ (𝐺 DProd 𝑇))) |
84 | 83 | simpld 475 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐺dom DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))) |
85 | 71, 77, 79, 80, 84 | dprdsplit 18447 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))) = ((𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))))) |
86 | 85 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))) = (#‘((𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})))))) |
87 | | eqid 2622 |
. . . . . . . . . . . . . . . 16
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
88 | | fdm 6051 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})):((𝑧 ∪ {𝑞}) ∖ {𝑃})⟶(SubGrp‘𝐺) → dom (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) = ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
89 | 71, 88 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → dom (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) = ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
90 | | ssdif 3745 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ⊆ (𝑧 ∪ {𝑞}) → (𝑧 ∖ {𝑃}) ⊆ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
91 | 60, 90 | mp1i 13 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑧 ∖ {𝑃}) ⊆ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
92 | 84, 89, 91 | dprdres 18427 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺dom DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})) ∧ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ⊆ (𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))) |
93 | 92 | simpld 475 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐺dom DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) |
94 | | dprdsubg 18423 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺dom DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ∈ (SubGrp‘𝐺)) |
95 | 93, 94 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ∈ (SubGrp‘𝐺)) |
96 | | ssun2 3777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑞} ⊆ (𝑧 ∪ {𝑞}) |
97 | | ssdif 3745 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑞} ⊆ (𝑧 ∪ {𝑞}) → ({𝑞} ∖ {𝑃}) ⊆ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
98 | 96, 97 | mp1i 13 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ({𝑞} ∖ {𝑃}) ⊆ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) |
99 | 84, 89, 98 | dprdres 18427 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺dom DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})) ∧ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ⊆ (𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))) |
100 | 99 | simpld 475 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐺dom DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) |
101 | | dprdsubg 18423 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺dom DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ∈ (SubGrp‘𝐺)) |
102 | 100, 101 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ∈ (SubGrp‘𝐺)) |
103 | 84, 89, 91, 98, 77, 48 | dprddisj2 18438 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ∩ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})))) = {(0g‘𝐺)}) |
104 | 84, 89, 91, 98, 77, 87 | dprdcntz2 18437 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))))) |
105 | | ablfac1.f |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐵 ∈ Fin) |
106 | 105 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐵 ∈ Fin) |
107 | | ablfac1.b |
. . . . . . . . . . . . . . . . . 18
⊢ 𝐵 = (Base‘𝐺) |
108 | 107 | dprdssv 18415 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ⊆ 𝐵 |
109 | | ssfi 8180 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ∈ Fin) |
110 | 106, 108,
109 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) ∈ Fin) |
111 | 107 | dprdssv 18415 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ⊆ 𝐵 |
112 | | ssfi 8180 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ⊆ 𝐵) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ∈ Fin) |
113 | 106, 111,
112 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) ∈ Fin) |
114 | 80, 48, 87, 95, 102, 103, 104, 110, 113 | lsmhash 18118 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘((𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})))(LSSum‘𝐺)(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))))) = ((#‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})))))) |
115 | 91 | resabs1d 5428 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})) = (𝑇 ↾ (𝑧 ∖ {𝑃}))) |
116 | 115 | oveq2d 6666 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃}))) = (𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) |
117 | 116 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})))) = (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) |
118 | 98 | resabs1d 5428 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})) = (𝑇 ↾ ({𝑞} ∖ {𝑃}))) |
119 | 118 | oveq2d 6666 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))) = (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) |
120 | 119 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃})))) = (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) |
121 | 117, 120 | oveq12d 6668 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((#‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd ((𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})) ↾ ({𝑞} ∖ {𝑃}))))) = ((#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))))) |
122 | 86, 114, 121 | 3eqtrd 2660 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))) = ((#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))))) |
123 | 122 | breq2d 4665 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))) ↔ 𝑃 ∥ ((#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))))) |
124 | 43 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝑃 ∈ ℙ) |
125 | 107 | dprdssv 18415 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))) ⊆ 𝐵 |
126 | | ssfi 8180 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))) ⊆ 𝐵) → (𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))) ∈ Fin) |
127 | 106, 125,
126 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))) ∈ Fin) |
128 | | hashcl 13147 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))) ∈ Fin → (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∈
ℕ0) |
129 | 127, 128 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∈
ℕ0) |
130 | 129 | nn0zd 11480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∈ ℤ) |
131 | 107 | dprdssv 18415 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) ⊆ 𝐵 |
132 | | ssfi 8180 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ Fin ∧ (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) ⊆ 𝐵) → (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) ∈ Fin) |
133 | 106, 131,
132 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) ∈ Fin) |
134 | | hashcl 13147 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) ∈ Fin → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) ∈
ℕ0) |
135 | 133, 134 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) ∈
ℕ0) |
136 | 135 | nn0zd 11480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) ∈ ℤ) |
137 | | euclemma 15425 |
. . . . . . . . . . . . . 14
⊢ ((𝑃 ∈ ℙ ∧
(#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∈ ℤ ∧ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) ∈ ℤ) → (𝑃 ∥ ((#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) ↔ (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∨ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))))) |
138 | 124, 130,
136, 137 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑃 ∥ ((#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) · (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) ↔ (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∨ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))))) |
139 | 123, 138 | bitrd 268 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))) ↔ (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∨ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))))) |
140 | 45 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → ¬ 𝑃 ∥ 1) |
141 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → 𝑞 = 𝑃) |
142 | 141 | sneqd 4189 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → {𝑞} = {𝑃}) |
143 | 142 | difeq1d 3727 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → ({𝑞} ∖ {𝑃}) = ({𝑃} ∖ {𝑃})) |
144 | | difid 3948 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ({𝑃} ∖ {𝑃}) = ∅ |
145 | 143, 144 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → ({𝑞} ∖ {𝑃}) = ∅) |
146 | 145 | reseq2d 5396 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (𝑇 ↾ ({𝑞} ∖ {𝑃})) = (𝑇 ↾ ∅)) |
147 | 146, 8 | syl6eq 2672 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (𝑇 ↾ ({𝑞} ∖ {𝑃})) = ∅) |
148 | 147 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) = (𝐺 DProd ∅)) |
149 | 51 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (𝐺 DProd ∅) =
{(0g‘𝐺)}) |
150 | 148, 149 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) = {(0g‘𝐺)}) |
151 | 150 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) = (#‘{(0g‘𝐺)})) |
152 | 151, 55 | syl6eq 2672 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) = 1) |
153 | 152 | breq2d 4665 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) ↔ 𝑃 ∥ 1)) |
154 | 140, 153 | mtbird 315 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 = 𝑃) → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) |
155 | | ablfac1.1 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐴 ⊆ ℙ) |
156 | 155 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐴 ⊆ ℙ) |
157 | 69 | unssbd 3791 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → {𝑞} ⊆ 𝐴) |
158 | | vex 3203 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑞 ∈ V |
159 | 158 | snss 4316 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑞 ∈ 𝐴 ↔ {𝑞} ⊆ 𝐴) |
160 | 157, 159 | sylibr 224 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝑞 ∈ 𝐴) |
161 | 156, 160 | sseldd 3604 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝑞 ∈ ℙ) |
162 | | ablfac1eu.3 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → 𝐶 ∈
ℕ0) |
163 | 160, 162 | syldan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → 𝐶 ∈
ℕ0) |
164 | | prmdvdsexpr 15429 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑃 ∈ ℙ ∧ 𝑞 ∈ ℙ ∧ 𝐶 ∈ ℕ0)
→ (𝑃 ∥ (𝑞↑𝐶) → 𝑃 = 𝑞)) |
165 | 124, 161,
163, 164 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑃 ∥ (𝑞↑𝐶) → 𝑃 = 𝑞)) |
166 | | eqcom 2629 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑃 = 𝑞 ↔ 𝑞 = 𝑃) |
167 | 165, 166 | syl6ib 241 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑃 ∥ (𝑞↑𝐶) → 𝑞 = 𝑃)) |
168 | 167 | necon3ad 2807 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑞 ≠ 𝑃 → ¬ 𝑃 ∥ (𝑞↑𝐶))) |
169 | 168 | imp 445 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → ¬ 𝑃 ∥ (𝑞↑𝐶)) |
170 | | disjsn2 4247 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑞 ≠ 𝑃 → ({𝑞} ∩ {𝑃}) = ∅) |
171 | 170 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → ({𝑞} ∩ {𝑃}) = ∅) |
172 | | disj3 4021 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (({𝑞} ∩ {𝑃}) = ∅ ↔ {𝑞} = ({𝑞} ∖ {𝑃})) |
173 | 171, 172 | sylib 208 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → {𝑞} = ({𝑞} ∖ {𝑃})) |
174 | 173 | reseq2d 5396 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (𝑇 ↾ {𝑞}) = (𝑇 ↾ ({𝑞} ∖ {𝑃}))) |
175 | 174 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (𝐺 DProd (𝑇 ↾ {𝑞})) = (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) |
176 | 65 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → 𝐺dom DProd 𝑇) |
177 | 66 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → dom 𝑇 = 𝐴) |
178 | 160 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → 𝑞 ∈ 𝐴) |
179 | 176, 177,
178 | dpjlem 18450 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (𝐺 DProd (𝑇 ↾ {𝑞})) = (𝑇‘𝑞)) |
180 | 175, 179 | eqtr3d 2658 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))) = (𝑇‘𝑞)) |
181 | 180 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) = (#‘(𝑇‘𝑞))) |
182 | | ablfac1eu.4 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴) → (#‘(𝑇‘𝑞)) = (𝑞↑𝐶)) |
183 | 160, 182 | syldan 487 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (#‘(𝑇‘𝑞)) = (𝑞↑𝐶)) |
184 | 183 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (#‘(𝑇‘𝑞)) = (𝑞↑𝐶)) |
185 | 181, 184 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) = (𝑞↑𝐶)) |
186 | 185 | breq2d 4665 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) ↔ 𝑃 ∥ (𝑞↑𝐶))) |
187 | 169, 186 | mtbird 315 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) ∧ 𝑞 ≠ 𝑃) → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) |
188 | 154, 187 | pm2.61dane 2881 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) |
189 | | orel2 398 |
. . . . . . . . . . . . 13
⊢ (¬
𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃})))) → ((𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∨ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) → 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) |
190 | 188, 189 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → ((𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) ∨ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ({𝑞} ∖ {𝑃}))))) → 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) |
191 | 139, 190 | sylbid 230 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))) → 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) |
192 | 191 | con3d 148 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (¬ 𝑞 ∈ 𝑧 ∧ (𝑧 ∪ {𝑞}) ⊆ 𝐴)) → (¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))) |
193 | 192 | expr 643 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝑞 ∈ 𝑧) → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → (¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))) → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))))) |
194 | 193 | a2d 29 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝑞 ∈ 𝑧) → (((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))))) |
195 | 63, 194 | syl5 34 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝑞 ∈ 𝑧) → ((𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃}))))))) |
196 | 195 | expcom 451 |
. . . . . 6
⊢ (¬
𝑞 ∈ 𝑧 → (𝜑 → ((𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))))) |
197 | 196 | adantl 482 |
. . . . 5
⊢ ((𝑧 ∈ Fin ∧ ¬ 𝑞 ∈ 𝑧) → (𝜑 → ((𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃}))))) → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))))) |
198 | 197 | a2d 29 |
. . . 4
⊢ ((𝑧 ∈ Fin ∧ ¬ 𝑞 ∈ 𝑧) → ((𝜑 → (𝑧 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝑧 ∖ {𝑃})))))) → (𝜑 → ((𝑧 ∪ {𝑞}) ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ ((𝑧 ∪ {𝑞}) ∖ {𝑃})))))))) |
199 | 15, 24, 33, 42, 59, 198 | findcard2s 8201 |
. . 3
⊢ (𝐴 ∈ Fin → (𝜑 → (𝐴 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃}))))))) |
200 | 2, 199 | mpcom 38 |
. 2
⊢ (𝜑 → (𝐴 ⊆ 𝐴 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃})))))) |
201 | 1, 200 | mpi 20 |
1
⊢ (𝜑 → ¬ 𝑃 ∥ (#‘(𝐺 DProd (𝑇 ↾ (𝐴 ∖ {𝑃}))))) |