Step |
Hyp |
Ref |
Expression |
1 |
|
psgnunilem2.g |
⊢ 𝐺 = ( SymGrp ‘ 𝐷 ) |
2 |
|
psgnunilem2.t |
⊢ 𝑇 = ran ( pmTrsp ‘ 𝐷 ) |
3 |
|
psgnunilem2.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑉 ) |
4 |
|
psgnunilem2.w |
⊢ ( 𝜑 → 𝑊 ∈ Word 𝑇 ) |
5 |
|
psgnunilem2.id |
⊢ ( 𝜑 → ( 𝐺 Σg 𝑊 ) = ( I ↾ 𝐷 ) ) |
6 |
|
psgnunilem2.l |
⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) = 𝐿 ) |
7 |
|
psgnunilem2.ix |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ 𝐿 ) ) |
8 |
|
psgnunilem2.a |
⊢ ( 𝜑 → 𝐴 ∈ dom ( ( 𝑊 ‘ 𝐼 ) ∖ I ) ) |
9 |
|
psgnunilem2.al |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑘 ) ∖ I ) ) |
10 |
|
noel |
⊢ ¬ 𝐴 ∈ ∅ |
11 |
5
|
difeq1d |
⊢ ( 𝜑 → ( ( 𝐺 Σg 𝑊 ) ∖ I ) = ( ( I ↾ 𝐷 ) ∖ I ) ) |
12 |
11
|
dmeqd |
⊢ ( 𝜑 → dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) = dom ( ( I ↾ 𝐷 ) ∖ I ) ) |
13 |
|
resss |
⊢ ( I ↾ 𝐷 ) ⊆ I |
14 |
|
ssdif0 |
⊢ ( ( I ↾ 𝐷 ) ⊆ I ↔ ( ( I ↾ 𝐷 ) ∖ I ) = ∅ ) |
15 |
13 14
|
mpbi |
⊢ ( ( I ↾ 𝐷 ) ∖ I ) = ∅ |
16 |
15
|
dmeqi |
⊢ dom ( ( I ↾ 𝐷 ) ∖ I ) = dom ∅ |
17 |
|
dm0 |
⊢ dom ∅ = ∅ |
18 |
16 17
|
eqtri |
⊢ dom ( ( I ↾ 𝐷 ) ∖ I ) = ∅ |
19 |
12 18
|
eqtrdi |
⊢ ( 𝜑 → dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) = ∅ ) |
20 |
19
|
eleq2d |
⊢ ( 𝜑 → ( 𝐴 ∈ dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) ↔ 𝐴 ∈ ∅ ) ) |
21 |
10 20
|
mtbiri |
⊢ ( 𝜑 → ¬ 𝐴 ∈ dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) ) |
22 |
1
|
symggrp |
⊢ ( 𝐷 ∈ 𝑉 → 𝐺 ∈ Grp ) |
23 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
24 |
3 22 23
|
3syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
26 |
2 1 25
|
symgtrf |
⊢ 𝑇 ⊆ ( Base ‘ 𝐺 ) |
27 |
|
sswrd |
⊢ ( 𝑇 ⊆ ( Base ‘ 𝐺 ) → Word 𝑇 ⊆ Word ( Base ‘ 𝐺 ) ) |
28 |
26 27
|
mp1i |
⊢ ( 𝜑 → Word 𝑇 ⊆ Word ( Base ‘ 𝐺 ) ) |
29 |
28 4
|
sseldd |
⊢ ( 𝜑 → 𝑊 ∈ Word ( Base ‘ 𝐺 ) ) |
30 |
|
pfxcl |
⊢ ( 𝑊 ∈ Word ( Base ‘ 𝐺 ) → ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) ) |
31 |
29 30
|
syl |
⊢ ( 𝜑 → ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) ) |
32 |
25
|
gsumwcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑊 prefix 𝐼 ) ∈ Word ( Base ‘ 𝐺 ) ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) ) |
33 |
24 31 32
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) ) |
34 |
1 25
|
symgbasf1o |
⊢ ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ ( Base ‘ 𝐺 ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷 –1-1-onto→ 𝐷 ) |
35 |
33 34
|
syl |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷 –1-1-onto→ 𝐷 ) |
36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) : 𝐷 –1-1-onto→ 𝐷 ) |
37 |
|
wrdf |
⊢ ( 𝑊 ∈ Word 𝑇 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑇 ) |
38 |
4 37
|
syl |
⊢ ( 𝜑 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑇 ) |
39 |
6
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝑊 ) ) = ( 0 ..^ 𝐿 ) ) |
40 |
7 39
|
eleqtrrd |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
41 |
38 40
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑊 ‘ 𝐼 ) ∈ 𝑇 ) |
42 |
26 41
|
sselid |
⊢ ( 𝜑 → ( 𝑊 ‘ 𝐼 ) ∈ ( Base ‘ 𝐺 ) ) |
43 |
1 25
|
symgbasf1o |
⊢ ( ( 𝑊 ‘ 𝐼 ) ∈ ( Base ‘ 𝐺 ) → ( 𝑊 ‘ 𝐼 ) : 𝐷 –1-1-onto→ 𝐷 ) |
44 |
42 43
|
syl |
⊢ ( 𝜑 → ( 𝑊 ‘ 𝐼 ) : 𝐷 –1-1-onto→ 𝐷 ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝑊 ‘ 𝐼 ) : 𝐷 –1-1-onto→ 𝐷 ) |
46 |
1 25
|
symgsssg |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubGrp ‘ 𝐺 ) ) |
47 |
|
subgsubm |
⊢ ( { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubGrp ‘ 𝐺 ) → { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubMnd ‘ 𝐺 ) ) |
48 |
3 46 47
|
3syl |
⊢ ( 𝜑 → { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ∈ ( SubMnd ‘ 𝐺 ) ) |
49 |
|
fzossfz |
⊢ ( 0 ..^ 𝐿 ) ⊆ ( 0 ... 𝐿 ) |
50 |
49 7
|
sselid |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ... 𝐿 ) ) |
51 |
6
|
oveq2d |
⊢ ( 𝜑 → ( 0 ... ( ♯ ‘ 𝑊 ) ) = ( 0 ... 𝐿 ) ) |
52 |
50 51
|
eleqtrrd |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) |
53 |
|
pfxmpt |
⊢ ( ( 𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ ( 0 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 prefix 𝐼 ) = ( 𝑠 ∈ ( 0 ..^ 𝐼 ) ↦ ( 𝑊 ‘ 𝑠 ) ) ) |
54 |
4 52 53
|
syl2anc |
⊢ ( 𝜑 → ( 𝑊 prefix 𝐼 ) = ( 𝑠 ∈ ( 0 ..^ 𝐼 ) ↦ ( 𝑊 ‘ 𝑠 ) ) ) |
55 |
|
difeq1 |
⊢ ( 𝑗 = ( 𝑊 ‘ 𝑠 ) → ( 𝑗 ∖ I ) = ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
56 |
55
|
dmeqd |
⊢ ( 𝑗 = ( 𝑊 ‘ 𝑠 ) → dom ( 𝑗 ∖ I ) = dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
57 |
56
|
sseq1d |
⊢ ( 𝑗 = ( 𝑊 ‘ 𝑠 ) → ( dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) ) |
58 |
|
disj2 |
⊢ ( ( dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) |
59 |
|
disjsn |
⊢ ( ( dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
60 |
58 59
|
bitr3i |
⊢ ( dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
61 |
57 60
|
bitrdi |
⊢ ( 𝑗 = ( 𝑊 ‘ 𝑠 ) → ( dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) ) |
62 |
|
elfzuz3 |
⊢ ( 𝐼 ∈ ( 0 ... 𝐿 ) → 𝐿 ∈ ( ℤ≥ ‘ 𝐼 ) ) |
63 |
50 62
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ ( ℤ≥ ‘ 𝐼 ) ) |
64 |
6 63
|
eqeltrd |
⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝐼 ) ) |
65 |
|
fzoss2 |
⊢ ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ≥ ‘ 𝐼 ) → ( 0 ..^ 𝐼 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
66 |
64 65
|
syl |
⊢ ( 𝜑 → ( 0 ..^ 𝐼 ) ⊆ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
67 |
66
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 ..^ 𝐼 ) ) → 𝑠 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) |
68 |
38
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 𝑠 ) ∈ 𝑇 ) |
69 |
26 68
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ 𝑠 ) ∈ ( Base ‘ 𝐺 ) ) |
70 |
67 69
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 ..^ 𝐼 ) ) → ( 𝑊 ‘ 𝑠 ) ∈ ( Base ‘ 𝐺 ) ) |
71 |
|
fveq2 |
⊢ ( 𝑘 = 𝑠 → ( 𝑊 ‘ 𝑘 ) = ( 𝑊 ‘ 𝑠 ) ) |
72 |
71
|
difeq1d |
⊢ ( 𝑘 = 𝑠 → ( ( 𝑊 ‘ 𝑘 ) ∖ I ) = ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
73 |
72
|
dmeqd |
⊢ ( 𝑘 = 𝑠 → dom ( ( 𝑊 ‘ 𝑘 ) ∖ I ) = dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
74 |
73
|
eleq2d |
⊢ ( 𝑘 = 𝑠 → ( 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑘 ) ∖ I ) ↔ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) ) |
75 |
74
|
notbid |
⊢ ( 𝑘 = 𝑠 → ( ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑘 ) ∖ I ) ↔ ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) ) |
76 |
75
|
cbvralvw |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑘 ) ∖ I ) ↔ ∀ 𝑠 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
77 |
9 76
|
sylib |
⊢ ( 𝜑 → ∀ 𝑠 ∈ ( 0 ..^ 𝐼 ) ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
78 |
77
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 ..^ 𝐼 ) ) → ¬ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝑠 ) ∖ I ) ) |
79 |
61 70 78
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 ..^ 𝐼 ) ) → ( 𝑊 ‘ 𝑠 ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ) |
80 |
54 79
|
fmpt3d |
⊢ ( 𝜑 → ( 𝑊 prefix 𝐼 ) : ( 0 ..^ 𝐼 ) ⟶ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ) |
81 |
80
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝑊 prefix 𝐼 ) : ( 0 ..^ 𝐼 ) ⟶ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ) |
82 |
|
iswrdi |
⊢ ( ( 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 } ) } ) |
83 |
81 82
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝑊 prefix 𝐼 ) ∈ Word { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ) |
84 |
|
gsumwsubmcl |
⊢ ( ( { 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 } ) } ) |
85 |
48 83 84
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } ) |
86 |
|
difeq1 |
⊢ ( 𝑗 = ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) → ( 𝑗 ∖ I ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ) |
87 |
86
|
dmeqd |
⊢ ( 𝑗 = ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) → dom ( 𝑗 ∖ I ) = dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ) |
88 |
87
|
sseq1d |
⊢ ( 𝑗 = ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) → ( dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) ) |
89 |
88
|
elrab |
⊢ ( ( 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 } ) ) ) |
90 |
89
|
simprbi |
⊢ ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } → dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) |
91 |
|
disj2 |
⊢ ( ( dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ) |
92 |
|
disjsn |
⊢ ( ( dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∩ { 𝐴 } ) = ∅ ↔ ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ) |
93 |
91 92
|
bitr3i |
⊢ ( dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊆ ( V ∖ { 𝐴 } ) ↔ ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ) |
94 |
90 93
|
sylib |
⊢ ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∈ { 𝑗 ∈ ( Base ‘ 𝐺 ) ∣ dom ( 𝑗 ∖ I ) ⊆ ( V ∖ { 𝐴 } ) } → ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ) |
95 |
85 94
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ) |
96 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝐴 ∈ dom ( ( 𝑊 ‘ 𝐼 ) ∖ I ) ) |
97 |
95 96
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ¬ 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ∧ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝐼 ) ∖ I ) ) ) |
98 |
97
|
olcd |
⊢ ( ( 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 |
⊢ ( ( 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 |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐴 ∈ dom ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∖ I ) ⊻ 𝐴 ∈ dom ( ( 𝑊 ‘ 𝐼 ) ∖ I ) ) ) |
101 |
|
f1omvdco3 |
⊢ ( ( ( 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 |
36 45 100 101
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝐴 ∈ dom ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊 ‘ 𝐼 ) ) ∖ I ) ) |
103 |
|
elfzo0 |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝐿 ) ↔ ( 𝐼 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿 ) ) |
104 |
103
|
simp2bi |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝐿 ) → 𝐿 ∈ ℕ ) |
105 |
7 104
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ ℕ ) |
106 |
6 105
|
eqeltrd |
⊢ ( 𝜑 → ( ♯ ‘ 𝑊 ) ∈ ℕ ) |
107 |
|
wrdfin |
⊢ ( 𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin ) |
108 |
|
hashnncl |
⊢ ( 𝑊 ∈ Fin → ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ 𝑊 ≠ ∅ ) ) |
109 |
4 107 108
|
3syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑊 ) ∈ ℕ ↔ 𝑊 ≠ ∅ ) ) |
110 |
106 109
|
mpbid |
⊢ ( 𝜑 → 𝑊 ≠ ∅ ) |
111 |
110
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝑊 ≠ ∅ ) |
112 |
|
pfxlswccat |
⊢ ( ( 𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅ ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ 〈“ ( lastS ‘ 𝑊 ) ”〉 ) = 𝑊 ) |
113 |
112
|
eqcomd |
⊢ ( ( 𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅ ) → 𝑊 = ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ 〈“ ( lastS ‘ 𝑊 ) ”〉 ) ) |
114 |
4 111 113
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝑊 = ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ 〈“ ( lastS ‘ 𝑊 ) ”〉 ) ) |
115 |
6
|
oveq1d |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( 𝐿 − 1 ) ) |
116 |
115
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = ( 𝐿 − 1 ) ) |
117 |
105
|
nncnd |
⊢ ( 𝜑 → 𝐿 ∈ ℂ ) |
118 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
119 |
|
elfzoelz |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝐿 ) → 𝐼 ∈ ℤ ) |
120 |
7 119
|
syl |
⊢ ( 𝜑 → 𝐼 ∈ ℤ ) |
121 |
120
|
zcnd |
⊢ ( 𝜑 → 𝐼 ∈ ℂ ) |
122 |
117 118 121
|
subadd2d |
⊢ ( 𝜑 → ( ( 𝐿 − 1 ) = 𝐼 ↔ ( 𝐼 + 1 ) = 𝐿 ) ) |
123 |
122
|
biimpar |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐿 − 1 ) = 𝐼 ) |
124 |
116 123
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) |
125 |
|
oveq2 |
⊢ ( ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 → ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 prefix 𝐼 ) ) |
126 |
125
|
adantl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 prefix 𝐼 ) ) |
127 |
|
lsw |
⊢ ( 𝑊 ∈ Word 𝑇 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
128 |
4 127
|
syl |
⊢ ( 𝜑 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ) |
129 |
|
fveq2 |
⊢ ( ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) = ( 𝑊 ‘ 𝐼 ) ) |
130 |
128 129
|
sylan9eq |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ 𝐼 ) ) |
131 |
130
|
s1eqd |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → 〈“ ( lastS ‘ 𝑊 ) ”〉 = 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) |
132 |
126 131
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) = 𝐼 ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ 〈“ ( lastS ‘ 𝑊 ) ”〉 ) = ( ( 𝑊 prefix 𝐼 ) ++ 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) |
133 |
124 132
|
syldan |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( 𝑊 prefix ( ( ♯ ‘ 𝑊 ) − 1 ) ) ++ 〈“ ( lastS ‘ 𝑊 ) ”〉 ) = ( ( 𝑊 prefix 𝐼 ) ++ 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) |
134 |
114 133
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝑊 = ( ( 𝑊 prefix 𝐼 ) ++ 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) |
135 |
134
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) ) |
136 |
42
|
s1cld |
⊢ ( 𝜑 → 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ∈ Word ( Base ‘ 𝐺 ) ) |
137 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
138 |
25 137
|
gsumccat |
⊢ ( ( 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 |
24 31 136 138
|
syl3anc |
⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) ) |
140 |
139
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg ( ( 𝑊 prefix 𝐼 ) ++ 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) ) |
141 |
25
|
gsumws1 |
⊢ ( ( 𝑊 ‘ 𝐼 ) ∈ ( Base ‘ 𝐺 ) → ( 𝐺 Σg 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) = ( 𝑊 ‘ 𝐼 ) ) |
142 |
42 141
|
syl |
⊢ ( 𝜑 → ( 𝐺 Σg 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) = ( 𝑊 ‘ 𝐼 ) ) |
143 |
142
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g ‘ 𝐺 ) ( 𝑊 ‘ 𝐼 ) ) ) |
144 |
1 25 137
|
symgov |
⊢ ( ( ( 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 |
33 42 144
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g ‘ 𝐺 ) ( 𝑊 ‘ 𝐼 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊 ‘ 𝐼 ) ) ) |
146 |
143 145
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg 〈“ ( 𝑊 ‘ 𝐼 ) ”〉 ) ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊 ‘ 𝐼 ) ) ) |
147 |
146
|
adantr |
⊢ ( ( 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 |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( 𝐺 Σg 𝑊 ) = ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊 ‘ 𝐼 ) ) ) |
149 |
148
|
difeq1d |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → ( ( 𝐺 Σg 𝑊 ) ∖ I ) = ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊 ‘ 𝐼 ) ) ∖ I ) ) |
150 |
149
|
dmeqd |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) = dom ( ( ( 𝐺 Σg ( 𝑊 prefix 𝐼 ) ) ∘ ( 𝑊 ‘ 𝐼 ) ) ∖ I ) ) |
151 |
102 150
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝐼 + 1 ) = 𝐿 ) → 𝐴 ∈ dom ( ( 𝐺 Σg 𝑊 ) ∖ I ) ) |
152 |
21 151
|
mtand |
⊢ ( 𝜑 → ¬ ( 𝐼 + 1 ) = 𝐿 ) |
153 |
|
fzostep1 |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝐿 ) → ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) ∨ ( 𝐼 + 1 ) = 𝐿 ) ) |
154 |
7 153
|
syl |
⊢ ( 𝜑 → ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) ∨ ( 𝐼 + 1 ) = 𝐿 ) ) |
155 |
154
|
ord |
⊢ ( 𝜑 → ( ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) → ( 𝐼 + 1 ) = 𝐿 ) ) |
156 |
152 155
|
mt3d |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝐿 ) ) |