Step | Hyp | Ref
| Expression |
1 | | noel 4178 |
. . . 4
⊢ ¬
𝐴 ∈
∅ |
2 | | psgnunilem2.id |
. . . . . . . 8
⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
3 | 2 | difeq1d 3984 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 Σg 𝑊) ∖ I ) = (( I ↾
𝐷) ∖ I
)) |
4 | 3 | dmeqd 5617 |
. . . . . 6
⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (( I ↾
𝐷) ∖ I
)) |
5 | | resss 5717 |
. . . . . . . . 9
⊢ ( I
↾ 𝐷) ⊆
I |
6 | | ssdif0 4204 |
. . . . . . . . 9
⊢ (( I
↾ 𝐷) ⊆ I ↔
(( I ↾ 𝐷) ∖ I )
= ∅) |
7 | 5, 6 | mpbi 222 |
. . . . . . . 8
⊢ (( I
↾ 𝐷) ∖ I ) =
∅ |
8 | 7 | dmeqi 5616 |
. . . . . . 7
⊢ dom (( I
↾ 𝐷) ∖ I ) =
dom ∅ |
9 | | dm0 5630 |
. . . . . . 7
⊢ dom
∅ = ∅ |
10 | 8, 9 | eqtri 2796 |
. . . . . 6
⊢ dom (( I
↾ 𝐷) ∖ I ) =
∅ |
11 | 4, 10 | syl6eq 2824 |
. . . . 5
⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) =
∅) |
12 | 11 | eleq2d 2845 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ) ↔ 𝐴 ∈
∅)) |
13 | 1, 12 | mtbiri 319 |
. . 3
⊢ (𝜑 → ¬ 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I )) |
14 | | psgnunilem2.d |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ 𝑉) |
15 | | psgnunilem2.g |
. . . . . . . . . 10
⊢ 𝐺 = (SymGrp‘𝐷) |
16 | 15 | symggrp 18279 |
. . . . . . . . 9
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
17 | | grpmnd 17888 |
. . . . . . . . 9
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
18 | 14, 16, 17 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
19 | | psgnunilem2.t |
. . . . . . . . . . . 12
⊢ 𝑇 = ran (pmTrsp‘𝐷) |
20 | | eqid 2772 |
. . . . . . . . . . . 12
⊢
(Base‘𝐺) =
(Base‘𝐺) |
21 | 19, 15, 20 | symgtrf 18348 |
. . . . . . . . . . 11
⊢ 𝑇 ⊆ (Base‘𝐺) |
22 | | sswrd 13670 |
. . . . . . . . . . 11
⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺)) |
23 | 21, 22 | mp1i 13 |
. . . . . . . . . 10
⊢ (𝜑 → Word 𝑇 ⊆ Word (Base‘𝐺)) |
24 | | psgnunilem2.w |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
25 | 23, 24 | sseldd 3855 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
26 | | pfxcl 13849 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) |
27 | 25, 26 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) |
28 | 20 | gsumwcl 17835 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ (Base‘𝐺)) |
29 | 18, 27, 28 | syl2anc 576 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ (Base‘𝐺)) |
30 | 15, 20 | symgbasf1o 18262 |
. . . . . . 7
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ (Base‘𝐺) → (𝐺 Σg (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷) |
31 | 29, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷) |
32 | 31 | adantr 473 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷) |
33 | | wrdf 13667 |
. . . . . . . . . 10
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
34 | 24, 33 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
35 | | psgnunilem2.ix |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) |
36 | | psgnunilem2.l |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) = 𝐿) |
37 | 36 | oveq2d 6986 |
. . . . . . . . . 10
⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿)) |
38 | 35, 37 | eleqtrrd 2863 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊))) |
39 | 34, 38 | ffvelrnd 6671 |
. . . . . . . 8
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
40 | 21, 39 | sseldi 3852 |
. . . . . . 7
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
41 | 15, 20 | symgbasf1o 18262 |
. . . . . . 7
⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
42 | 40, 41 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
43 | 42 | adantr 473 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
44 | 15, 20 | symgsssg 18346 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑉 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺)) |
45 | | subgsubm 18075 |
. . . . . . . . . . 11
⊢ ({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubGrp‘𝐺) →
{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubMnd‘𝐺)) |
46 | 14, 44, 45 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺)) |
47 | | fzossfz 12865 |
. . . . . . . . . . . . . . . 16
⊢
(0..^𝐿) ⊆
(0...𝐿) |
48 | 47, 35 | sseldi 3852 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
49 | 36 | oveq2d 6986 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿)) |
50 | 48, 49 | eleqtrrd 2863 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ (0...(♯‘𝑊))) |
51 | | pfxmpt 13850 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐼) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
52 | 24, 50, 51 | syl2anc 576 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑊 prefix 𝐼) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
53 | | difeq1 3978 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑊‘𝑠) → (𝑗 ∖ I ) = ((𝑊‘𝑠) ∖ I )) |
54 | 53 | dmeqd 5617 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑊‘𝑠) → dom (𝑗 ∖ I ) = dom ((𝑊‘𝑠) ∖ I )) |
55 | 54 | sseq1d 3884 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}))) |
56 | | disj2 4284 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴})) |
57 | | disjsn 4515 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
58 | 56, 57 | bitr3i 269 |
. . . . . . . . . . . . . . 15
⊢ (dom
((𝑊‘𝑠) ∖ I ) ⊆ (V ∖
{𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
59 | 55, 58 | syl6bb 279 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
60 | | elfzuz3 12714 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐼)) |
61 | 48, 60 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐿 ∈ (ℤ≥‘𝐼)) |
62 | 36, 61 | eqeltrd 2860 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (♯‘𝑊) ∈
(ℤ≥‘𝐼)) |
63 | | fzoss2 12873 |
. . . . . . . . . . . . . . . . 17
⊢
((♯‘𝑊)
∈ (ℤ≥‘𝐼) → (0..^𝐼) ⊆ (0..^(♯‘𝑊))) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0..^𝐼) ⊆ (0..^(♯‘𝑊))) |
65 | 64 | sselda 3854 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → 𝑠 ∈ (0..^(♯‘𝑊))) |
66 | 34 | ffvelrnda 6670 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ 𝑇) |
67 | 21, 66 | sseldi 3852 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ (Base‘𝐺)) |
68 | 65, 67 | syldan 582 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ (Base‘𝐺)) |
69 | | psgnunilem2.al |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I )) |
70 | | fveq2 6493 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑠 → (𝑊‘𝑘) = (𝑊‘𝑠)) |
71 | 70 | difeq1d 3984 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑠 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑠) ∖ I )) |
72 | 71 | dmeqd 5617 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑠 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑠) ∖ I )) |
73 | 72 | eleq2d 2845 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑠 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
74 | 73 | notbid 310 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑠 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
75 | 74 | cbvralv 3377 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑘 ∈
(0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
76 | 69, 75 | sylib 210 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
77 | 76 | r19.21bi 3152 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
78 | 59, 68, 77 | elrabd 3592 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
79 | 52, 78 | fmpt3d 6697 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
80 | 79 | adantr 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
81 | | iswrdi 13666 |
. . . . . . . . . . 11
⊢ ((𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
82 | 80, 81 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
83 | | gsumwsubmcl 17833 |
. . . . . . . . . 10
⊢ (({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubMnd‘𝐺) ∧
(𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
84 | 46, 82, 83 | syl2an2r 672 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
85 | | difeq1 3978 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝐺 Σg (𝑊 prefix 𝐼)) → (𝑗 ∖ I ) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
86 | 85 | dmeqd 5617 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐺 Σg (𝑊 prefix 𝐼)) → dom (𝑗 ∖ I ) = dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
87 | 86 | sseq1d 3884 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝐺 Σg (𝑊 prefix 𝐼)) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝐺 Σg
(𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}))) |
88 | 87 | elrab 3589 |
. . . . . . . . . . 11
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ ((𝐺 Σg (𝑊 prefix 𝐼)) ∈ (Base‘𝐺) ∧ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}))) |
89 | 88 | simprbi 489 |
. . . . . . . . . 10
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → dom ((𝐺 Σg
(𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴})) |
90 | | disj2 4284 |
. . . . . . . . . . 11
⊢ ((dom
((𝐺
Σg (𝑊 prefix 𝐼)) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴})) |
91 | | disjsn 4515 |
. . . . . . . . . . 11
⊢ ((dom
((𝐺
Σg (𝑊 prefix 𝐼)) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
92 | 90, 91 | bitr3i 269 |
. . . . . . . . . 10
⊢ (dom
((𝐺
Σg (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
93 | 89, 92 | sylib 210 |
. . . . . . . . 9
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
94 | 84, 93 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
95 | | psgnunilem2.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
96 | 95 | adantr 473 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
97 | 94, 96 | jca 504 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) |
98 | 97 | olcd 860 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))) |
99 | | excxor 1493 |
. . . . . 6
⊢ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ↔ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))) |
100 | 98, 99 | sylibr 226 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) |
101 | | f1omvdco3 18328 |
. . . . 5
⊢ (((𝐺 Σg
(𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷 ∧ (𝑊‘𝐼):𝐷–1-1-onto→𝐷 ∧ (𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
102 | 32, 43, 100, 101 | syl3anc 1351 |
. . . 4
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
103 | | elfzo0 12886 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ (0..^𝐿) ↔ (𝐼 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿)) |
104 | 103 | simp2bi 1126 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∈ (0..^𝐿) → 𝐿 ∈ ℕ) |
105 | 35, 104 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℕ) |
106 | 36, 105 | eqeltrd 2860 |
. . . . . . . . . . . 12
⊢ (𝜑 → (♯‘𝑊) ∈
ℕ) |
107 | | wrdfin 13683 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin) |
108 | | hashnncl 13535 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Fin →
((♯‘𝑊) ∈
ℕ ↔ 𝑊 ≠
∅)) |
109 | 24, 107, 108 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅)) |
110 | 106, 109 | mpbid 224 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ≠ ∅) |
111 | 110 | adantr 473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ≠ ∅) |
112 | | pfxlswccat 13892 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉) = 𝑊) |
113 | 112 | eqcomd 2778 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → 𝑊 = ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉)) |
114 | 24, 111, 113 | syl2an2r 672 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉)) |
115 | 36 | oveq1d 6985 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((♯‘𝑊) − 1) = (𝐿 − 1)) |
116 | 115 | adantr 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = (𝐿 − 1)) |
117 | 105 | nncnd 11449 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℂ) |
118 | | 1cnd 10426 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℂ) |
119 | | elfzoelz 12847 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ) |
120 | 35, 119 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ ℤ) |
121 | 120 | zcnd 11894 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ ℂ) |
122 | 117, 118,
121 | subadd2d 10809 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐿 − 1) = 𝐼 ↔ (𝐼 + 1) = 𝐿)) |
123 | 122 | biimpar 470 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐿 − 1) = 𝐼) |
124 | 116, 123 | eqtrd 2808 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = 𝐼) |
125 | | oveq2 6978 |
. . . . . . . . . . . 12
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
(𝑊 prefix
((♯‘𝑊) −
1)) = (𝑊 prefix 𝐼)) |
126 | 125 | adantl 474 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (𝑊 prefix ((♯‘𝑊) − 1)) = (𝑊 prefix 𝐼)) |
127 | | lsw 13717 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ Word 𝑇 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
128 | 24, 127 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
129 | | fveq2 6493 |
. . . . . . . . . . . . 13
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
(𝑊‘((♯‘𝑊) − 1)) = (𝑊‘𝐼)) |
130 | 128, 129 | sylan9eq 2828 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (lastS‘𝑊) = (𝑊‘𝐼)) |
131 | 130 | s1eqd 13754 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → 〈“(lastS‘𝑊)”〉 =
〈“(𝑊‘𝐼)”〉) |
132 | 126, 131 | oveq12d 6988 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉) = ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) |
133 | 124, 132 | syldan 582 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉) = ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) |
134 | 114, 133 | eqtrd 2808 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) |
135 | 134 | oveq2d 6986 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉))) |
136 | 40 | s1cld 13756 |
. . . . . . . . 9
⊢ (𝜑 → 〈“(𝑊‘𝐼)”〉 ∈ Word (Base‘𝐺)) |
137 | | eqid 2772 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
138 | 20, 137 | gsumccat 17836 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺) ∧ 〈“(𝑊‘𝐼)”〉 ∈ Word (Base‘𝐺)) → (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
139 | 18, 27, 136, 138 | syl3anc 1351 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
140 | 139 | adantr 473 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
141 | 20 | gsumws1 17834 |
. . . . . . . . . . 11
⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝐺 Σg
〈“(𝑊‘𝐼)”〉) = (𝑊‘𝐼)) |
142 | 40, 141 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)”〉) = (𝑊‘𝐼)) |
143 | 142 | oveq2d 6986 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 prefix 𝐼))(+g‘𝐺)(𝑊‘𝐼))) |
144 | 15, 20, 137 | symgov 18269 |
. . . . . . . . . 10
⊢ (((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ (Base‘𝐺) ∧ (𝑊‘𝐼) ∈ (Base‘𝐺)) → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
145 | 29, 40, 144 | syl2anc 576 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
146 | 143, 145 | eqtrd 2808 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
147 | 146 | adantr 473 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
148 | 135, 140,
147 | 3eqtrd 2812 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
149 | 148 | difeq1d 3984 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg 𝑊) ∖ I ) = (((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
150 | 149 | dmeqd 5617 |
. . . 4
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
151 | 102, 150 | eleqtrrd 2863 |
. . 3
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I )) |
152 | 13, 151 | mtand 803 |
. 2
⊢ (𝜑 → ¬ (𝐼 + 1) = 𝐿) |
153 | | fzostep1 12961 |
. . . 4
⊢ (𝐼 ∈ (0..^𝐿) → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿)) |
154 | 35, 153 | syl 17 |
. . 3
⊢ (𝜑 → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿)) |
155 | 154 | ord 850 |
. 2
⊢ (𝜑 → (¬ (𝐼 + 1) ∈ (0..^𝐿) → (𝐼 + 1) = 𝐿)) |
156 | 152, 155 | mt3d 143 |
1
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |