Step | Hyp | Ref
| Expression |
1 | | noel 4067 |
. . . 4
⊢ ¬
𝐴 ∈
∅ |
2 | | psgnunilem2.id |
. . . . . . . 8
⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
3 | 2 | difeq1d 3878 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 Σg 𝑊) ∖ I ) = (( I ↾
𝐷) ∖ I
)) |
4 | 3 | dmeqd 5462 |
. . . . . 6
⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (( I ↾
𝐷) ∖ I
)) |
5 | | resss 5561 |
. . . . . . . . 9
⊢ ( I
↾ 𝐷) ⊆
I |
6 | | ssdif0 4089 |
. . . . . . . . 9
⊢ (( I
↾ 𝐷) ⊆ I ↔
(( I ↾ 𝐷) ∖ I )
= ∅) |
7 | 5, 6 | mpbi 220 |
. . . . . . . 8
⊢ (( I
↾ 𝐷) ∖ I ) =
∅ |
8 | 7 | dmeqi 5461 |
. . . . . . 7
⊢ dom (( I
↾ 𝐷) ∖ I ) =
dom ∅ |
9 | | dm0 5475 |
. . . . . . 7
⊢ dom
∅ = ∅ |
10 | 8, 9 | eqtri 2793 |
. . . . . 6
⊢ dom (( I
↾ 𝐷) ∖ I ) =
∅ |
11 | 4, 10 | syl6eq 2821 |
. . . . 5
⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) =
∅) |
12 | 11 | eleq2d 2836 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ) ↔ 𝐴 ∈
∅)) |
13 | 1, 12 | mtbiri 316 |
. . 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 18020 |
. . . . . . . . 9
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
17 | | grpmnd 17630 |
. . . . . . . . 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 2771 |
. . . . . . . . . . . 12
⊢
(Base‘𝐺) =
(Base‘𝐺) |
21 | 19, 15, 20 | symgtrf 18089 |
. . . . . . . . . . 11
⊢ 𝑇 ⊆ (Base‘𝐺) |
22 | | sswrd 13502 |
. . . . . . . . . . 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 3753 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
26 | | swrdcl 13620 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 substr 〈0, 𝐼〉) ∈ Word (Base‘𝐺)) |
27 | 25, 26 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 substr 〈0, 𝐼〉) ∈ Word (Base‘𝐺)) |
28 | 20 | gsumwcl 17578 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑊 substr 〈0, 𝐼〉) ∈ Word
(Base‘𝐺)) →
(𝐺
Σg (𝑊 substr 〈0, 𝐼〉)) ∈ (Base‘𝐺)) |
29 | 18, 27, 28 | syl2anc 573 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∈
(Base‘𝐺)) |
30 | 15, 20 | symgbasf1o 18003 |
. . . . . . 7
⊢ ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈
(Base‘𝐺) →
(𝐺
Σg (𝑊 substr 〈0, 𝐼〉)):𝐷–1-1-onto→𝐷) |
31 | 29, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)):𝐷–1-1-onto→𝐷) |
32 | 31 | adantr 466 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)):𝐷–1-1-onto→𝐷) |
33 | | wrdf 13499 |
. . . . . . . . . 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 6807 |
. . . . . . . . . 10
⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿)) |
38 | 35, 37 | eleqtrrd 2853 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊))) |
39 | 34, 38 | ffvelrnd 6501 |
. . . . . . . 8
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
40 | 21, 39 | sseldi 3750 |
. . . . . . 7
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
41 | 15, 20 | symgbasf1o 18003 |
. . . . . . 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 466 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
44 | 15, 20 | symgsssg 18087 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ 𝑉 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺)) |
45 | | subgsubm 17817 |
. . . . . . . . . . . 12
⊢ ({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 |
. . . . . . . . . . 11
⊢ (𝜑 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺)) |
47 | 46 | adantr 466 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺)) |
48 | | fzossfz 12689 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(0..^𝐿) ⊆
(0...𝐿) |
49 | 48, 35 | sseldi 3750 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
50 | | elfzuz3 12539 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐼)) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐿 ∈ (ℤ≥‘𝐼)) |
52 | 36, 51 | eqeltrd 2850 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (♯‘𝑊) ∈
(ℤ≥‘𝐼)) |
53 | | fzoss2 12697 |
. . . . . . . . . . . . . . . . . 18
⊢
((♯‘𝑊)
∈ (ℤ≥‘𝐼) → (0..^𝐼) ⊆ (0..^(♯‘𝑊))) |
54 | 52, 53 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (0..^𝐼) ⊆ (0..^(♯‘𝑊))) |
55 | 54 | sselda 3752 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → 𝑠 ∈ (0..^(♯‘𝑊))) |
56 | 34 | ffvelrnda 6500 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ 𝑇) |
57 | 21, 56 | sseldi 3750 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ (Base‘𝐺)) |
58 | 55, 57 | syldan 579 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ (Base‘𝐺)) |
59 | | psgnunilem2.al |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I )) |
60 | | fveq2 6330 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑠 → (𝑊‘𝑘) = (𝑊‘𝑠)) |
61 | 60 | difeq1d 3878 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑠 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑠) ∖ I )) |
62 | 61 | dmeqd 5462 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑠 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑠) ∖ I )) |
63 | 62 | eleq2d 2836 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑠 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
64 | 63 | notbid 307 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑠 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
65 | 64 | cbvralv 3320 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑘 ∈
(0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
66 | 59, 65 | sylib 208 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
67 | 66 | r19.21bi 3081 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
68 | | difeq1 3872 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑊‘𝑠) → (𝑗 ∖ I ) = ((𝑊‘𝑠) ∖ I )) |
69 | 68 | dmeqd 5462 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑊‘𝑠) → dom (𝑗 ∖ I ) = dom ((𝑊‘𝑠) ∖ I )) |
70 | 69 | sseq1d 3781 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}))) |
71 | | disj2 4168 |
. . . . . . . . . . . . . . . . . 18
⊢ ((dom
((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴})) |
72 | | disjsn 4383 |
. . . . . . . . . . . . . . . . . 18
⊢ ((dom
((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
73 | 71, 72 | bitr3i 266 |
. . . . . . . . . . . . . . . . 17
⊢ (dom
((𝑊‘𝑠) ∖ I ) ⊆ (V ∖
{𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
74 | 70, 73 | syl6bb 276 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
75 | 74 | elrab 3515 |
. . . . . . . . . . . . . . 15
⊢ ((𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ ((𝑊‘𝑠) ∈ (Base‘𝐺) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
76 | 58, 67, 75 | sylanbrc 572 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
77 | | eqid 2771 |
. . . . . . . . . . . . . 14
⊢ (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)) |
78 | 76, 77 | fmptd 6525 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
79 | 36 | oveq2d 6807 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿)) |
80 | 49, 79 | eleqtrrd 2853 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐼 ∈ (0...(♯‘𝑊))) |
81 | | swrd0val 13622 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ (0...(♯‘𝑊))) → (𝑊 substr 〈0, 𝐼〉) = (𝑊 ↾ (0..^𝐼))) |
82 | 24, 80, 81 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑊 substr 〈0, 𝐼〉) = (𝑊 ↾ (0..^𝐼))) |
83 | 34 | feqmptd 6389 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑊 = (𝑠 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘𝑠))) |
84 | 83 | reseq1d 5531 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑊 ↾ (0..^𝐼)) = ((𝑠 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘𝑠)) ↾ (0..^𝐼))) |
85 | | resmpt 5588 |
. . . . . . . . . . . . . . . 16
⊢
((0..^𝐼) ⊆
(0..^(♯‘𝑊))
→ ((𝑠 ∈
(0..^(♯‘𝑊))
↦ (𝑊‘𝑠)) ↾ (0..^𝐼)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
86 | 52, 53, 85 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑠 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘𝑠)) ↾ (0..^𝐼)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
87 | 82, 84, 86 | 3eqtrd 2809 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑊 substr 〈0, 𝐼〉) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
88 | 87 | feq1d 6168 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑊 substr 〈0, 𝐼〉):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})) |
89 | 78, 88 | mpbird 247 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊 substr 〈0, 𝐼〉):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
90 | 89 | adantr 466 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 substr 〈0, 𝐼〉):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
91 | | iswrdi 13498 |
. . . . . . . . . . 11
⊢ ((𝑊 substr 〈0, 𝐼〉):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → (𝑊 substr 〈0, 𝐼〉) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
92 | 90, 91 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 substr 〈0, 𝐼〉) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
93 | | gsumwsubmcl 17576 |
. . . . . . . . . 10
⊢ (({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubMnd‘𝐺) ∧
(𝑊 substr 〈0, 𝐼〉) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})}) → (𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})}) |
94 | 47, 92, 93 | syl2anc 573 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})}) |
95 | | difeq1 3872 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) → (𝑗 ∖ I ) = ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∖ I
)) |
96 | 95 | dmeqd 5462 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) → dom (𝑗 ∖ I ) = dom ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∖ I
)) |
97 | 96 | sseq1d 3781 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) → (dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴}) ↔ dom ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊆ (V
∖ {𝐴}))) |
98 | 97 | elrab 3515 |
. . . . . . . . . . 11
⊢ ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ↔ ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈
(Base‘𝐺) ∧ dom
((𝐺
Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊆ (V ∖
{𝐴}))) |
99 | 98 | simprbi 484 |
. . . . . . . . . 10
⊢ ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} → dom ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊆ (V
∖ {𝐴})) |
100 | | disj2 4168 |
. . . . . . . . . . 11
⊢ ((dom
((𝐺
Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊆ (V
∖ {𝐴})) |
101 | | disjsn 4383 |
. . . . . . . . . . 11
⊢ ((dom
((𝐺
Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I
)) |
102 | 100, 101 | bitr3i 266 |
. . . . . . . . . 10
⊢ (dom
((𝐺
Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊆ (V ∖
{𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I
)) |
103 | 99, 102 | sylib 208 |
. . . . . . . . 9
⊢ ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I
)) |
104 | 94, 103 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I
)) |
105 | | psgnunilem2.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
106 | 105 | adantr 466 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
107 | 104, 106 | jca 501 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) |
108 | 107 | olcd 863 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∧ ¬
𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))) |
109 | | excxor 1617 |
. . . . . 6
⊢ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊻
𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ↔ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∧ ¬
𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))) |
110 | 108, 109 | sylibr 224 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊻
𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) |
111 | | f1omvdco3 18069 |
. . . . 5
⊢ (((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)):𝐷–1-1-onto→𝐷 ∧ (𝑊‘𝐼):𝐷–1-1-onto→𝐷 ∧ (𝐴 ∈ dom ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∖ I ) ⊻
𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼)) ∖ I )) |
112 | 32, 43, 110, 111 | syl3anc 1476 |
. . . 4
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼)) ∖ I )) |
113 | 24 | adantr 466 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ∈ Word 𝑇) |
114 | | elfzo0 12710 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ (0..^𝐿) ↔ (𝐼 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿)) |
115 | 114 | simp2bi 1140 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∈ (0..^𝐿) → 𝐿 ∈ ℕ) |
116 | 35, 115 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℕ) |
117 | 36, 116 | eqeltrd 2850 |
. . . . . . . . . . . 12
⊢ (𝜑 → (♯‘𝑊) ∈
ℕ) |
118 | | wrdfin 13512 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin) |
119 | | hashnncl 13352 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Fin →
((♯‘𝑊) ∈
ℕ ↔ 𝑊 ≠
∅)) |
120 | 24, 118, 119 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅)) |
121 | 117, 120 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ≠ ∅) |
122 | 121 | adantr 466 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ≠ ∅) |
123 | | swrdccatwrd 13670 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → ((𝑊 substr 〈0, ((♯‘𝑊) − 1)〉) ++
〈“(lastS‘𝑊)”〉) = 𝑊) |
124 | 123 | eqcomd 2777 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → 𝑊 = ((𝑊 substr 〈0, ((♯‘𝑊) − 1)〉) ++
〈“(lastS‘𝑊)”〉)) |
125 | 113, 122,
124 | syl2anc 573 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 substr 〈0, ((♯‘𝑊) − 1)〉) ++
〈“(lastS‘𝑊)”〉)) |
126 | 36 | oveq1d 6806 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((♯‘𝑊) − 1) = (𝐿 − 1)) |
127 | 126 | adantr 466 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = (𝐿 − 1)) |
128 | 116 | nncnd 11236 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℂ) |
129 | | 1cnd 10256 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℂ) |
130 | | elfzoelz 12671 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ) |
131 | 35, 130 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ ℤ) |
132 | 131 | zcnd 11683 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ ℂ) |
133 | 128, 129,
132 | subadd2d 10611 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐿 − 1) = 𝐼 ↔ (𝐼 + 1) = 𝐿)) |
134 | 133 | biimpar 463 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐿 − 1) = 𝐼) |
135 | 127, 134 | eqtrd 2805 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = 𝐼) |
136 | | opeq2 4540 |
. . . . . . . . . . . . 13
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
〈0, ((♯‘𝑊)
− 1)〉 = 〈0, 𝐼〉) |
137 | 136 | oveq2d 6807 |
. . . . . . . . . . . 12
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
(𝑊 substr 〈0,
((♯‘𝑊) −
1)〉) = (𝑊 substr
〈0, 𝐼〉)) |
138 | 137 | adantl 467 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (𝑊 substr 〈0, ((♯‘𝑊) − 1)〉) = (𝑊 substr 〈0, 𝐼〉)) |
139 | | lsw 13541 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ Word 𝑇 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
140 | 24, 139 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
141 | | fveq2 6330 |
. . . . . . . . . . . . 13
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
(𝑊‘((♯‘𝑊) − 1)) = (𝑊‘𝐼)) |
142 | 140, 141 | sylan9eq 2825 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (lastS‘𝑊) = (𝑊‘𝐼)) |
143 | 142 | s1eqd 13574 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → 〈“(lastS‘𝑊)”〉 =
〈“(𝑊‘𝐼)”〉) |
144 | 138, 143 | oveq12d 6809 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → ((𝑊 substr 〈0, ((♯‘𝑊) − 1)〉) ++
〈“(lastS‘𝑊)”〉) = ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉)) |
145 | 135, 144 | syldan 579 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝑊 substr 〈0, ((♯‘𝑊) − 1)〉) ++
〈“(lastS‘𝑊)”〉) = ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉)) |
146 | 125, 145 | eqtrd 2805 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉)) |
147 | 146 | oveq2d 6807 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = (𝐺 Σg ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉))) |
148 | 40 | s1cld 13576 |
. . . . . . . . 9
⊢ (𝜑 → 〈“(𝑊‘𝐼)”〉 ∈ Word (Base‘𝐺)) |
149 | | eqid 2771 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
150 | 20, 149 | gsumccat 17579 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ (𝑊 substr 〈0, 𝐼〉) ∈ Word
(Base‘𝐺) ∧
〈“(𝑊‘𝐼)”〉 ∈ Word
(Base‘𝐺)) →
(𝐺
Σg ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
151 | 18, 27, 148, 150 | syl3anc 1476 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 Σg ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
152 | 151 | adantr 466 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg ((𝑊 substr 〈0, 𝐼〉) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
153 | 20 | gsumws1 17577 |
. . . . . . . . . . 11
⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝐺 Σg
〈“(𝑊‘𝐼)”〉) = (𝑊‘𝐼)) |
154 | 40, 153 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)”〉) = (𝑊‘𝐼)) |
155 | 154 | oveq2d 6807 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝑊‘𝐼))) |
156 | 15, 20, 149 | symgov 18010 |
. . . . . . . . . 10
⊢ (((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∈
(Base‘𝐺) ∧ (𝑊‘𝐼) ∈ (Base‘𝐺)) → ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼))) |
157 | 29, 40, 156 | syl2anc 573 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼))) |
158 | 155, 157 | eqtrd 2805 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼))) |
159 | 158 | adantr 466 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼))) |
160 | 147, 152,
159 | 3eqtrd 2809 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = ((𝐺 Σg (𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼))) |
161 | 160 | difeq1d 3878 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg 𝑊) ∖ I ) = (((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼)) ∖ I )) |
162 | 161 | dmeqd 5462 |
. . . 4
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (((𝐺 Σg
(𝑊 substr 〈0, 𝐼〉)) ∘ (𝑊‘𝐼)) ∖ I )) |
163 | 112, 162 | eleqtrrd 2853 |
. . 3
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I )) |
164 | 13, 163 | mtand 817 |
. 2
⊢ (𝜑 → ¬ (𝐼 + 1) = 𝐿) |
165 | | fzostep1 12785 |
. . . 4
⊢ (𝐼 ∈ (0..^𝐿) → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿)) |
166 | 35, 165 | syl 17 |
. . 3
⊢ (𝜑 → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿)) |
167 | 166 | ord 853 |
. 2
⊢ (𝜑 → (¬ (𝐼 + 1) ∈ (0..^𝐿) → (𝐼 + 1) = 𝐿)) |
168 | 164, 167 | mt3d 142 |
1
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |