Step | Hyp | Ref
| Expression |
1 | | psgnunilem3.l |
. . . 4
⊢ (𝜑 → (#‘𝑊) = 𝐿) |
2 | | psgnunilem3.w2 |
. . . 4
⊢ (𝜑 → (#‘𝑊) ∈ ℕ) |
3 | 1, 2 | eqeltrrd 2689 |
. . 3
⊢ (𝜑 → 𝐿 ∈ ℕ) |
4 | 3 | nnnn0d 11228 |
. 2
⊢ (𝜑 → 𝐿 ∈
ℕ0) |
5 | | psgnunilem3.w1 |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
6 | | wrdf 13165 |
. . . . . . 7
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(#‘𝑊))⟶𝑇) |
7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑊:(0..^(#‘𝑊))⟶𝑇) |
8 | | 0nn0 11184 |
. . . . . . . . 9
⊢ 0 ∈
ℕ0 |
9 | 8 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℕ0) |
10 | 3 | nngt0d 10941 |
. . . . . . . 8
⊢ (𝜑 → 0 < 𝐿) |
11 | | elfzo0 12376 |
. . . . . . . 8
⊢ (0 ∈
(0..^𝐿) ↔ (0 ∈
ℕ0 ∧ 𝐿
∈ ℕ ∧ 0 < 𝐿)) |
12 | 9, 3, 10, 11 | syl3anbrc 1239 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ (0..^𝐿)) |
13 | 1 | oveq2d 6565 |
. . . . . . 7
⊢ (𝜑 → (0..^(#‘𝑊)) = (0..^𝐿)) |
14 | 12, 13 | eleqtrrd 2691 |
. . . . . 6
⊢ (𝜑 → 0 ∈
(0..^(#‘𝑊))) |
15 | 7, 14 | ffvelrnd 6268 |
. . . . 5
⊢ (𝜑 → (𝑊‘0) ∈ 𝑇) |
16 | | eqid 2610 |
. . . . . 6
⊢
(pmTrsp‘𝐷) =
(pmTrsp‘𝐷) |
17 | | psgnunilem3.t |
. . . . . 6
⊢ 𝑇 = ran (pmTrsp‘𝐷) |
18 | 16, 17 | pmtrfmvdn0 17705 |
. . . . 5
⊢ ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠
∅) |
19 | 15, 18 | syl 17 |
. . . 4
⊢ (𝜑 → dom ((𝑊‘0) ∖ I ) ≠
∅) |
20 | | n0 3890 |
. . . 4
⊢ (dom
((𝑊‘0) ∖ I )
≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I )) |
21 | 19, 20 | sylib 207 |
. . 3
⊢ (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I )) |
22 | | fzonel 12352 |
. . . . . . . 8
⊢ ¬
𝐿 ∈ (0..^𝐿) |
23 | | simpr1 1060 |
. . . . . . . 8
⊢ ((((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) → 𝐿 ∈ (0..^𝐿)) |
24 | 22, 23 | mto 187 |
. . . . . . 7
⊢ ¬
(((𝐺
Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) |
25 | 24 | a1i 11 |
. . . . . 6
⊢ (𝑤 ∈ Word 𝑇 → ¬ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
26 | 25 | nrex 2983 |
. . . . 5
⊢ ¬
∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) |
27 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿))) |
28 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 0 → (𝑤‘𝑎) = (𝑤‘0)) |
29 | 28 | difeq1d 3689 |
. . . . . . . . . . . 12
⊢ (𝑎 = 0 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘0) ∖ I )) |
30 | 29 | dmeqd 5248 |
. . . . . . . . . . 11
⊢ (𝑎 = 0 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I )) |
31 | 30 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝑎 = 0 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I ))) |
32 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑎 = 0 → (0..^𝑎) = (0..^0)) |
33 | 32 | raleqdv 3121 |
. . . . . . . . . 10
⊢ (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) |
34 | 27, 31, 33 | 3anbi123d 1391 |
. . . . . . . . 9
⊢ (𝑎 = 0 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
35 | 34 | anbi2d 736 |
. . . . . . . 8
⊢ (𝑎 = 0 → ((((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
36 | 35 | rexbidv 3034 |
. . . . . . 7
⊢ (𝑎 = 0 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
37 | 36 | imbi2d 329 |
. . . . . 6
⊢ (𝑎 = 0 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))) |
38 | | eleq1 2676 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿))) |
39 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑏 → (𝑤‘𝑎) = (𝑤‘𝑏)) |
40 | 39 | difeq1d 3689 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑏 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘𝑏) ∖ I )) |
41 | 40 | dmeqd 5248 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘𝑏) ∖ I )) |
42 | 41 | eleq2d 2673 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ))) |
43 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏)) |
44 | 43 | raleqdv 3121 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) |
45 | 38, 42, 44 | 3anbi123d 1391 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
46 | 45 | anbi2d 736 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
47 | 46 | rexbidv 3034 |
. . . . . . . 8
⊢ (𝑎 = 𝑏 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
48 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥)) |
49 | 48 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
50 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥)) |
51 | 50 | eqeq1d 2612 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑥 → ((#‘𝑤) = 𝐿 ↔ (#‘𝑥) = 𝐿)) |
52 | 49, 51 | anbi12d 743 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿))) |
53 | | fveq1 6102 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑥 → (𝑤‘𝑏) = (𝑥‘𝑏)) |
54 | 53 | difeq1d 3689 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑥 → ((𝑤‘𝑏) ∖ I ) = ((𝑥‘𝑏) ∖ I )) |
55 | 54 | dmeqd 5248 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑥 → dom ((𝑤‘𝑏) ∖ I ) = dom ((𝑥‘𝑏) ∖ I )) |
56 | 55 | eleq2d 2673 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ))) |
57 | | fveq1 6102 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 𝑥 → (𝑤‘𝑐) = (𝑥‘𝑐)) |
58 | 57 | difeq1d 3689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑥 → ((𝑤‘𝑐) ∖ I ) = ((𝑥‘𝑐) ∖ I )) |
59 | 58 | dmeqd 5248 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑥 → dom ((𝑤‘𝑐) ∖ I ) = dom ((𝑥‘𝑐) ∖ I )) |
60 | 59 | eleq2d 2673 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ))) |
61 | 60 | notbid 307 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ))) |
62 | 61 | ralbidv 2969 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ))) |
63 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = 𝑑 → (𝑥‘𝑐) = (𝑥‘𝑑)) |
64 | 63 | difeq1d 3689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = 𝑑 → ((𝑥‘𝑐) ∖ I ) = ((𝑥‘𝑑) ∖ I )) |
65 | 64 | dmeqd 5248 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 = 𝑑 → dom ((𝑥‘𝑐) ∖ I ) = dom ((𝑥‘𝑑) ∖ I )) |
66 | 65 | eleq2d 2673 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) |
67 | 66 | notbid 307 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) |
68 | 67 | cbvralv 3147 |
. . . . . . . . . . . 12
⊢
(∀𝑐 ∈
(0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )) |
69 | 62, 68 | syl6bb 275 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) |
70 | 56, 69 | 3anbi23d 1394 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑥 → ((𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))) |
71 | 52, 70 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝑤 = 𝑥 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) |
72 | 71 | cbvrexv 3148 |
. . . . . . . 8
⊢
(∃𝑤 ∈
Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))) |
73 | 47, 72 | syl6bb 275 |
. . . . . . 7
⊢ (𝑎 = 𝑏 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) |
74 | 73 | imbi2d 329 |
. . . . . 6
⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))))) |
75 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿))) |
76 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑏 + 1) → (𝑤‘𝑎) = (𝑤‘(𝑏 + 1))) |
77 | 76 | difeq1d 3689 |
. . . . . . . . . . . 12
⊢ (𝑎 = (𝑏 + 1) → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I )) |
78 | 77 | dmeqd 5248 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑏 + 1) → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I )) |
79 | 78 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ))) |
80 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1))) |
81 | 80 | raleqdv 3121 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑏 + 1) → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) |
82 | 75, 79, 81 | 3anbi123d 1391 |
. . . . . . . . 9
⊢ (𝑎 = (𝑏 + 1) → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
83 | 82 | anbi2d 736 |
. . . . . . . 8
⊢ (𝑎 = (𝑏 + 1) → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
84 | 83 | rexbidv 3034 |
. . . . . . 7
⊢ (𝑎 = (𝑏 + 1) → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
85 | 84 | imbi2d 329 |
. . . . . 6
⊢ (𝑎 = (𝑏 + 1) → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))) |
86 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿))) |
87 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝐿 → (𝑤‘𝑎) = (𝑤‘𝐿)) |
88 | 87 | difeq1d 3689 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐿 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘𝐿) ∖ I )) |
89 | 88 | dmeqd 5248 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐿 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘𝐿) ∖ I )) |
90 | 89 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ))) |
91 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿)) |
92 | 91 | raleqdv 3121 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) |
93 | 86, 90, 92 | 3anbi123d 1391 |
. . . . . . . . 9
⊢ (𝑎 = 𝐿 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
94 | 93 | anbi2d 736 |
. . . . . . . 8
⊢ (𝑎 = 𝐿 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
95 | 94 | rexbidv 3034 |
. . . . . . 7
⊢ (𝑎 = 𝐿 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
96 | 95 | imbi2d 329 |
. . . . . 6
⊢ (𝑎 = 𝐿 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))) |
97 | 5 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑊 ∈ Word 𝑇) |
98 | | psgnunilem3.w3 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
99 | 98, 1 | jca 553 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (#‘𝑊) = 𝐿)) |
100 | 99 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ((𝐺 Σg
𝑊) = ( I ↾ 𝐷) ∧ (#‘𝑊) = 𝐿)) |
101 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 0 ∈
(0..^𝐿)) |
102 | | simpr 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑒 ∈ dom ((𝑊‘0) ∖ I )) |
103 | | ral0 4028 |
. . . . . . . . . 10
⊢
∀𝑐 ∈
∅ ¬ 𝑒 ∈ dom
((𝑊‘𝑐) ∖ I ) |
104 | | fzo0 12361 |
. . . . . . . . . . 11
⊢ (0..^0) =
∅ |
105 | 104 | raleqi 3119 |
. . . . . . . . . 10
⊢
(∀𝑐 ∈
(0..^0) ¬ 𝑒 ∈ dom
((𝑊‘𝑐) ∖ I ) ↔
∀𝑐 ∈ ∅
¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )) |
106 | 103, 105 | mpbir 220 |
. . . . . . . . 9
⊢
∀𝑐 ∈
(0..^0) ¬ 𝑒 ∈ dom
((𝑊‘𝑐) ∖ I ) |
107 | 106 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )) |
108 | 101, 102,
107 | 3jca 1235 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈
(0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))) |
109 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊)) |
110 | 109 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))) |
111 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊)) |
112 | 111 | eqeq1d 2612 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → ((#‘𝑤) = 𝐿 ↔ (#‘𝑊) = 𝐿)) |
113 | 110, 112 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (#‘𝑊) = 𝐿))) |
114 | | fveq1 6102 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0)) |
115 | 114 | difeq1d 3689 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I )) |
116 | 115 | dmeqd 5248 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I
)) |
117 | 116 | eleq2d 2673 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I ))) |
118 | | fveq1 6102 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑊 → (𝑤‘𝑐) = (𝑊‘𝑐)) |
119 | 118 | difeq1d 3689 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑊 → ((𝑤‘𝑐) ∖ I ) = ((𝑊‘𝑐) ∖ I )) |
120 | 119 | dmeqd 5248 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑊 → dom ((𝑤‘𝑐) ∖ I ) = dom ((𝑊‘𝑐) ∖ I )) |
121 | 120 | eleq2d 2673 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))) |
122 | 121 | notbid 307 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))) |
123 | 122 | ralbidv 2969 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))) |
124 | 117, 123 | 3anbi23d 1394 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))) |
125 | 113, 124 | anbi12d 743 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (#‘𝑊) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))))) |
126 | 125 | rspcev 3282 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (#‘𝑊) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
127 | 97, 100, 108, 126 | syl12anc 1316 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
128 | | psgnunilem3.g |
. . . . . . . . . 10
⊢ 𝐺 = (SymGrp‘𝐷) |
129 | | psgnunilem3.d |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ 𝑉) |
130 | 129 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝐷 ∈ 𝑉) |
131 | | simprl 790 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝑥 ∈ Word 𝑇) |
132 | | simpll 786 |
. . . . . . . . . . 11
⊢ ((((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) |
133 | 132 | ad2antll 761 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) |
134 | | simplr 788 |
. . . . . . . . . . 11
⊢ ((((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → (#‘𝑥) = 𝐿) |
135 | 134 | ad2antll 761 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → (#‘𝑥) = 𝐿) |
136 | | simpr1 1060 |
. . . . . . . . . . 11
⊢ ((((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → 𝑏 ∈ (0..^𝐿)) |
137 | 136 | ad2antll 761 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝑏 ∈ (0..^𝐿)) |
138 | | simpr2 1061 |
. . . . . . . . . . 11
⊢ ((((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I )) |
139 | 138 | ad2antll 761 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I )) |
140 | | simpr3 1062 |
. . . . . . . . . . 11
⊢ ((((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )) |
141 | 140 | ad2antll 761 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )) |
142 | | psgnunilem3.in |
. . . . . . . . . . . 12
⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
143 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦)) |
144 | 143 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → ((#‘𝑥) = (𝐿 − 2) ↔ (#‘𝑦) = (𝐿 − 2))) |
145 | | oveq2 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦)) |
146 | 145 | eqeq1d 2612 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) |
147 | 144, 146 | anbi12d 743 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((#‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))) |
148 | 147 | cbvrexv 3148 |
. . . . . . . . . . . 12
⊢
(∃𝑥 ∈
Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) |
149 | 142, 148 | sylnib 317 |
. . . . . . . . . . 11
⊢ (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) |
150 | 149 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) |
151 | 128, 17, 130, 131, 133, 135, 137, 139, 141, 150 | psgnunilem2 17738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) |
152 | 151 | rexlimdvaa 3014 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
153 | 152 | a2i 14 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))) → ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
154 | 153 | a1i 11 |
. . . . . 6
⊢ (𝑏 ∈ ℕ0
→ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (#‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))) → ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))) |
155 | 37, 74, 85, 96, 127, 154 | nn0ind 11348 |
. . . . 5
⊢ (𝐿 ∈ ℕ0
→ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))) |
156 | 26, 155 | mtoi 189 |
. . . 4
⊢ (𝐿 ∈ ℕ0
→ ¬ (𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ))) |
157 | 156 | con2i 133 |
. . 3
⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈
ℕ0) |
158 | 21, 157 | exlimddv 1850 |
. 2
⊢ (𝜑 → ¬ 𝐿 ∈
ℕ0) |
159 | 4, 158 | pm2.65i 184 |
1
⊢ ¬
𝜑 |