Step |
Hyp |
Ref |
Expression |
1 |
|
psgnunilem2.g |
|- G = ( SymGrp ` D ) |
2 |
|
psgnunilem2.t |
|- T = ran ( pmTrsp ` D ) |
3 |
|
psgnunilem2.d |
|- ( ph -> D e. V ) |
4 |
|
psgnunilem2.w |
|- ( ph -> W e. Word T ) |
5 |
|
psgnunilem2.id |
|- ( ph -> ( G gsum W ) = ( _I |` D ) ) |
6 |
|
psgnunilem2.l |
|- ( ph -> ( # ` W ) = L ) |
7 |
|
psgnunilem2.ix |
|- ( ph -> I e. ( 0 ..^ L ) ) |
8 |
|
psgnunilem2.a |
|- ( ph -> A e. dom ( ( W ` I ) \ _I ) ) |
9 |
|
psgnunilem2.al |
|- ( ph -> A. k e. ( 0 ..^ I ) -. A e. dom ( ( W ` k ) \ _I ) ) |
10 |
|
psgnunilem2.in |
|- ( ph -> -. E. x e. Word T ( ( # ` x ) = ( L - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) |
11 |
|
wrd0 |
|- (/) e. Word T |
12 |
|
splcl |
|- ( ( W e. Word T /\ (/) e. Word T ) -> ( W splice <. I , ( I + 2 ) , (/) >. ) e. Word T ) |
13 |
4 11 12
|
sylancl |
|- ( ph -> ( W splice <. I , ( I + 2 ) , (/) >. ) e. Word T ) |
14 |
13
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( W splice <. I , ( I + 2 ) , (/) >. ) e. Word T ) |
15 |
|
fzossfz |
|- ( 0 ..^ L ) C_ ( 0 ... L ) |
16 |
15 7
|
sseldi |
|- ( ph -> I e. ( 0 ... L ) ) |
17 |
|
elfznn0 |
|- ( I e. ( 0 ... L ) -> I e. NN0 ) |
18 |
16 17
|
syl |
|- ( ph -> I e. NN0 ) |
19 |
|
2nn0 |
|- 2 e. NN0 |
20 |
|
nn0addcl |
|- ( ( I e. NN0 /\ 2 e. NN0 ) -> ( I + 2 ) e. NN0 ) |
21 |
18 19 20
|
sylancl |
|- ( ph -> ( I + 2 ) e. NN0 ) |
22 |
18
|
nn0red |
|- ( ph -> I e. RR ) |
23 |
|
nn0addge1 |
|- ( ( I e. RR /\ 2 e. NN0 ) -> I <_ ( I + 2 ) ) |
24 |
22 19 23
|
sylancl |
|- ( ph -> I <_ ( I + 2 ) ) |
25 |
|
elfz2nn0 |
|- ( I e. ( 0 ... ( I + 2 ) ) <-> ( I e. NN0 /\ ( I + 2 ) e. NN0 /\ I <_ ( I + 2 ) ) ) |
26 |
18 21 24 25
|
syl3anbrc |
|- ( ph -> I e. ( 0 ... ( I + 2 ) ) ) |
27 |
1 2 3 4 5 6 7 8 9
|
psgnunilem5 |
|- ( ph -> ( I + 1 ) e. ( 0 ..^ L ) ) |
28 |
|
fzofzp1 |
|- ( ( I + 1 ) e. ( 0 ..^ L ) -> ( ( I + 1 ) + 1 ) e. ( 0 ... L ) ) |
29 |
27 28
|
syl |
|- ( ph -> ( ( I + 1 ) + 1 ) e. ( 0 ... L ) ) |
30 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
31 |
30
|
oveq2i |
|- ( I + 2 ) = ( I + ( 1 + 1 ) ) |
32 |
18
|
nn0cnd |
|- ( ph -> I e. CC ) |
33 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
34 |
32 33 33
|
addassd |
|- ( ph -> ( ( I + 1 ) + 1 ) = ( I + ( 1 + 1 ) ) ) |
35 |
31 34
|
eqtr4id |
|- ( ph -> ( I + 2 ) = ( ( I + 1 ) + 1 ) ) |
36 |
6
|
oveq2d |
|- ( ph -> ( 0 ... ( # ` W ) ) = ( 0 ... L ) ) |
37 |
29 35 36
|
3eltr4d |
|- ( ph -> ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) |
38 |
11
|
a1i |
|- ( ph -> (/) e. Word T ) |
39 |
4 26 37 38
|
spllen |
|- ( ph -> ( # ` ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( ( # ` W ) + ( ( # ` (/) ) - ( ( I + 2 ) - I ) ) ) ) |
40 |
|
hash0 |
|- ( # ` (/) ) = 0 |
41 |
40
|
oveq1i |
|- ( ( # ` (/) ) - ( ( I + 2 ) - I ) ) = ( 0 - ( ( I + 2 ) - I ) ) |
42 |
|
df-neg |
|- -u ( ( I + 2 ) - I ) = ( 0 - ( ( I + 2 ) - I ) ) |
43 |
41 42
|
eqtr4i |
|- ( ( # ` (/) ) - ( ( I + 2 ) - I ) ) = -u ( ( I + 2 ) - I ) |
44 |
|
2cn |
|- 2 e. CC |
45 |
|
pncan2 |
|- ( ( I e. CC /\ 2 e. CC ) -> ( ( I + 2 ) - I ) = 2 ) |
46 |
32 44 45
|
sylancl |
|- ( ph -> ( ( I + 2 ) - I ) = 2 ) |
47 |
46
|
negeqd |
|- ( ph -> -u ( ( I + 2 ) - I ) = -u 2 ) |
48 |
43 47
|
syl5eq |
|- ( ph -> ( ( # ` (/) ) - ( ( I + 2 ) - I ) ) = -u 2 ) |
49 |
6 48
|
oveq12d |
|- ( ph -> ( ( # ` W ) + ( ( # ` (/) ) - ( ( I + 2 ) - I ) ) ) = ( L + -u 2 ) ) |
50 |
|
elfzel2 |
|- ( I e. ( 0 ... L ) -> L e. ZZ ) |
51 |
16 50
|
syl |
|- ( ph -> L e. ZZ ) |
52 |
51
|
zcnd |
|- ( ph -> L e. CC ) |
53 |
|
negsub |
|- ( ( L e. CC /\ 2 e. CC ) -> ( L + -u 2 ) = ( L - 2 ) ) |
54 |
52 44 53
|
sylancl |
|- ( ph -> ( L + -u 2 ) = ( L - 2 ) ) |
55 |
39 49 54
|
3eqtrd |
|- ( ph -> ( # ` ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( L - 2 ) ) |
56 |
55
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( # ` ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( L - 2 ) ) |
57 |
|
splid |
|- ( ( W e. Word T /\ ( I e. ( 0 ... ( I + 2 ) ) /\ ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) ) -> ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) = W ) |
58 |
4 26 37 57
|
syl12anc |
|- ( ph -> ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) = W ) |
59 |
58
|
oveq2d |
|- ( ph -> ( G gsum ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) ) = ( G gsum W ) ) |
60 |
59
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( G gsum ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) ) = ( G gsum W ) ) |
61 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
62 |
1
|
symggrp |
|- ( D e. V -> G e. Grp ) |
63 |
3 62
|
syl |
|- ( ph -> G e. Grp ) |
64 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
65 |
63 64
|
syl |
|- ( ph -> G e. Mnd ) |
66 |
65
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> G e. Mnd ) |
67 |
2 1 61
|
symgtrf |
|- T C_ ( Base ` G ) |
68 |
|
sswrd |
|- ( T C_ ( Base ` G ) -> Word T C_ Word ( Base ` G ) ) |
69 |
67 68
|
ax-mp |
|- Word T C_ Word ( Base ` G ) |
70 |
69 4
|
sseldi |
|- ( ph -> W e. Word ( Base ` G ) ) |
71 |
70
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> W e. Word ( Base ` G ) ) |
72 |
26
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> I e. ( 0 ... ( I + 2 ) ) ) |
73 |
37
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) |
74 |
|
swrdcl |
|- ( W e. Word ( Base ` G ) -> ( W substr <. I , ( I + 2 ) >. ) e. Word ( Base ` G ) ) |
75 |
70 74
|
syl |
|- ( ph -> ( W substr <. I , ( I + 2 ) >. ) e. Word ( Base ` G ) ) |
76 |
75
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( W substr <. I , ( I + 2 ) >. ) e. Word ( Base ` G ) ) |
77 |
|
wrd0 |
|- (/) e. Word ( Base ` G ) |
78 |
77
|
a1i |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> (/) e. Word ( Base ` G ) ) |
79 |
6
|
oveq2d |
|- ( ph -> ( 0 ..^ ( # ` W ) ) = ( 0 ..^ L ) ) |
80 |
27 79
|
eleqtrrd |
|- ( ph -> ( I + 1 ) e. ( 0 ..^ ( # ` W ) ) ) |
81 |
|
swrds2 |
|- ( ( W e. Word T /\ I e. NN0 /\ ( I + 1 ) e. ( 0 ..^ ( # ` W ) ) ) -> ( W substr <. I , ( I + 2 ) >. ) = <" ( W ` I ) ( W ` ( I + 1 ) ) "> ) |
82 |
4 18 80 81
|
syl3anc |
|- ( ph -> ( W substr <. I , ( I + 2 ) >. ) = <" ( W ` I ) ( W ` ( I + 1 ) ) "> ) |
83 |
82
|
oveq2d |
|- ( ph -> ( G gsum ( W substr <. I , ( I + 2 ) >. ) ) = ( G gsum <" ( W ` I ) ( W ` ( I + 1 ) ) "> ) ) |
84 |
|
wrdf |
|- ( W e. Word T -> W : ( 0 ..^ ( # ` W ) ) --> T ) |
85 |
4 84
|
syl |
|- ( ph -> W : ( 0 ..^ ( # ` W ) ) --> T ) |
86 |
79
|
feq2d |
|- ( ph -> ( W : ( 0 ..^ ( # ` W ) ) --> T <-> W : ( 0 ..^ L ) --> T ) ) |
87 |
85 86
|
mpbid |
|- ( ph -> W : ( 0 ..^ L ) --> T ) |
88 |
87 7
|
ffvelrnd |
|- ( ph -> ( W ` I ) e. T ) |
89 |
67 88
|
sseldi |
|- ( ph -> ( W ` I ) e. ( Base ` G ) ) |
90 |
87 27
|
ffvelrnd |
|- ( ph -> ( W ` ( I + 1 ) ) e. T ) |
91 |
67 90
|
sseldi |
|- ( ph -> ( W ` ( I + 1 ) ) e. ( Base ` G ) ) |
92 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
93 |
61 92
|
gsumws2 |
|- ( ( G e. Mnd /\ ( W ` I ) e. ( Base ` G ) /\ ( W ` ( I + 1 ) ) e. ( Base ` G ) ) -> ( G gsum <" ( W ` I ) ( W ` ( I + 1 ) ) "> ) = ( ( W ` I ) ( +g ` G ) ( W ` ( I + 1 ) ) ) ) |
94 |
65 89 91 93
|
syl3anc |
|- ( ph -> ( G gsum <" ( W ` I ) ( W ` ( I + 1 ) ) "> ) = ( ( W ` I ) ( +g ` G ) ( W ` ( I + 1 ) ) ) ) |
95 |
1 61 92
|
symgov |
|- ( ( ( W ` I ) e. ( Base ` G ) /\ ( W ` ( I + 1 ) ) e. ( Base ` G ) ) -> ( ( W ` I ) ( +g ` G ) ( W ` ( I + 1 ) ) ) = ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) ) |
96 |
89 91 95
|
syl2anc |
|- ( ph -> ( ( W ` I ) ( +g ` G ) ( W ` ( I + 1 ) ) ) = ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) ) |
97 |
83 94 96
|
3eqtrd |
|- ( ph -> ( G gsum ( W substr <. I , ( I + 2 ) >. ) ) = ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) ) |
98 |
97
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( G gsum ( W substr <. I , ( I + 2 ) >. ) ) = ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) ) |
99 |
|
simpr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) |
100 |
1
|
symgid |
|- ( D e. V -> ( _I |` D ) = ( 0g ` G ) ) |
101 |
3 100
|
syl |
|- ( ph -> ( _I |` D ) = ( 0g ` G ) ) |
102 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
103 |
102
|
gsum0 |
|- ( G gsum (/) ) = ( 0g ` G ) |
104 |
101 103
|
eqtr4di |
|- ( ph -> ( _I |` D ) = ( G gsum (/) ) ) |
105 |
104
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( _I |` D ) = ( G gsum (/) ) ) |
106 |
98 99 105
|
3eqtrd |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( G gsum ( W substr <. I , ( I + 2 ) >. ) ) = ( G gsum (/) ) ) |
107 |
61 66 71 72 73 76 78 106
|
gsumspl |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( G gsum ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) ) = ( G gsum ( W splice <. I , ( I + 2 ) , (/) >. ) ) ) |
108 |
5
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( G gsum W ) = ( _I |` D ) ) |
109 |
60 107 108
|
3eqtr3d |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> ( G gsum ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( _I |` D ) ) |
110 |
|
fveqeq2 |
|- ( x = ( W splice <. I , ( I + 2 ) , (/) >. ) -> ( ( # ` x ) = ( L - 2 ) <-> ( # ` ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( L - 2 ) ) ) |
111 |
|
oveq2 |
|- ( x = ( W splice <. I , ( I + 2 ) , (/) >. ) -> ( G gsum x ) = ( G gsum ( W splice <. I , ( I + 2 ) , (/) >. ) ) ) |
112 |
111
|
eqeq1d |
|- ( x = ( W splice <. I , ( I + 2 ) , (/) >. ) -> ( ( G gsum x ) = ( _I |` D ) <-> ( G gsum ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( _I |` D ) ) ) |
113 |
110 112
|
anbi12d |
|- ( x = ( W splice <. I , ( I + 2 ) , (/) >. ) -> ( ( ( # ` x ) = ( L - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) <-> ( ( # ` ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( L - 2 ) /\ ( G gsum ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( _I |` D ) ) ) ) |
114 |
113
|
rspcev |
|- ( ( ( W splice <. I , ( I + 2 ) , (/) >. ) e. Word T /\ ( ( # ` ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( L - 2 ) /\ ( G gsum ( W splice <. I , ( I + 2 ) , (/) >. ) ) = ( _I |` D ) ) ) -> E. x e. Word T ( ( # ` x ) = ( L - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) |
115 |
14 56 109 114
|
syl12anc |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> E. x e. Word T ( ( # ` x ) = ( L - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) |
116 |
10
|
adantr |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> -. E. x e. Word T ( ( # ` x ) = ( L - 2 ) /\ ( G gsum x ) = ( _I |` D ) ) ) |
117 |
115 116
|
pm2.21dd |
|- ( ( ph /\ ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) ) -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) |
118 |
117
|
ex |
|- ( ph -> ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) ) |
119 |
4
|
adantr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> W e. Word T ) |
120 |
|
simprl |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> r e. T ) |
121 |
|
simprr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> s e. T ) |
122 |
120 121
|
s2cld |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> <" r s "> e. Word T ) |
123 |
|
splcl |
|- ( ( W e. Word T /\ <" r s "> e. Word T ) -> ( W splice <. I , ( I + 2 ) , <" r s "> >. ) e. Word T ) |
124 |
119 122 123
|
syl2anc |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( W splice <. I , ( I + 2 ) , <" r s "> >. ) e. Word T ) |
125 |
124
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( W splice <. I , ( I + 2 ) , <" r s "> >. ) e. Word T ) |
126 |
65
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> G e. Mnd ) |
127 |
70
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> W e. Word ( Base ` G ) ) |
128 |
26
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> I e. ( 0 ... ( I + 2 ) ) ) |
129 |
37
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) |
130 |
69 122
|
sseldi |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> <" r s "> e. Word ( Base ` G ) ) |
131 |
130
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> <" r s "> e. Word ( Base ` G ) ) |
132 |
75
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( W substr <. I , ( I + 2 ) >. ) e. Word ( Base ` G ) ) |
133 |
|
simprr1 |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) ) |
134 |
97
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum ( W substr <. I , ( I + 2 ) >. ) ) = ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) ) |
135 |
65
|
adantr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> G e. Mnd ) |
136 |
67
|
a1i |
|- ( ph -> T C_ ( Base ` G ) ) |
137 |
136
|
sselda |
|- ( ( ph /\ r e. T ) -> r e. ( Base ` G ) ) |
138 |
137
|
adantrr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> r e. ( Base ` G ) ) |
139 |
136
|
sselda |
|- ( ( ph /\ s e. T ) -> s e. ( Base ` G ) ) |
140 |
139
|
adantrl |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> s e. ( Base ` G ) ) |
141 |
61 92
|
gsumws2 |
|- ( ( G e. Mnd /\ r e. ( Base ` G ) /\ s e. ( Base ` G ) ) -> ( G gsum <" r s "> ) = ( r ( +g ` G ) s ) ) |
142 |
135 138 140 141
|
syl3anc |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( G gsum <" r s "> ) = ( r ( +g ` G ) s ) ) |
143 |
1 61 92
|
symgov |
|- ( ( r e. ( Base ` G ) /\ s e. ( Base ` G ) ) -> ( r ( +g ` G ) s ) = ( r o. s ) ) |
144 |
138 140 143
|
syl2anc |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( r ( +g ` G ) s ) = ( r o. s ) ) |
145 |
142 144
|
eqtrd |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( G gsum <" r s "> ) = ( r o. s ) ) |
146 |
145
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum <" r s "> ) = ( r o. s ) ) |
147 |
133 134 146
|
3eqtr4rd |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum <" r s "> ) = ( G gsum ( W substr <. I , ( I + 2 ) >. ) ) ) |
148 |
61 126 127 128 129 131 132 147
|
gsumspl |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( G gsum ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) ) ) |
149 |
59
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum ( W splice <. I , ( I + 2 ) , ( W substr <. I , ( I + 2 ) >. ) >. ) ) = ( G gsum W ) ) |
150 |
5
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum W ) = ( _I |` D ) ) |
151 |
148 149 150
|
3eqtrd |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( _I |` D ) ) |
152 |
26
|
adantr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> I e. ( 0 ... ( I + 2 ) ) ) |
153 |
37
|
adantr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) |
154 |
119 152 153 122
|
spllen |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( ( # ` W ) + ( ( # ` <" r s "> ) - ( ( I + 2 ) - I ) ) ) ) |
155 |
|
s2len |
|- ( # ` <" r s "> ) = 2 |
156 |
155
|
oveq1i |
|- ( ( # ` <" r s "> ) - ( ( I + 2 ) - I ) ) = ( 2 - ( ( I + 2 ) - I ) ) |
157 |
46
|
oveq2d |
|- ( ph -> ( 2 - ( ( I + 2 ) - I ) ) = ( 2 - 2 ) ) |
158 |
44
|
subidi |
|- ( 2 - 2 ) = 0 |
159 |
157 158
|
eqtrdi |
|- ( ph -> ( 2 - ( ( I + 2 ) - I ) ) = 0 ) |
160 |
156 159
|
syl5eq |
|- ( ph -> ( ( # ` <" r s "> ) - ( ( I + 2 ) - I ) ) = 0 ) |
161 |
160
|
oveq2d |
|- ( ph -> ( ( # ` W ) + ( ( # ` <" r s "> ) - ( ( I + 2 ) - I ) ) ) = ( ( # ` W ) + 0 ) ) |
162 |
6 52
|
eqeltrd |
|- ( ph -> ( # ` W ) e. CC ) |
163 |
162
|
addid1d |
|- ( ph -> ( ( # ` W ) + 0 ) = ( # ` W ) ) |
164 |
161 163 6
|
3eqtrd |
|- ( ph -> ( ( # ` W ) + ( ( # ` <" r s "> ) - ( ( I + 2 ) - I ) ) ) = L ) |
165 |
164
|
adantr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( # ` W ) + ( ( # ` <" r s "> ) - ( ( I + 2 ) - I ) ) ) = L ) |
166 |
154 165
|
eqtrd |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) |
167 |
166
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) |
168 |
151 167
|
jca |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( _I |` D ) /\ ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) ) |
169 |
27
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( I + 1 ) e. ( 0 ..^ L ) ) |
170 |
|
simprr2 |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> A e. dom ( s \ _I ) ) |
171 |
|
1nn0 |
|- 1 e. NN0 |
172 |
|
2nn |
|- 2 e. NN |
173 |
|
1lt2 |
|- 1 < 2 |
174 |
|
elfzo0 |
|- ( 1 e. ( 0 ..^ 2 ) <-> ( 1 e. NN0 /\ 2 e. NN /\ 1 < 2 ) ) |
175 |
171 172 173 174
|
mpbir3an |
|- 1 e. ( 0 ..^ 2 ) |
176 |
155
|
oveq2i |
|- ( 0 ..^ ( # ` <" r s "> ) ) = ( 0 ..^ 2 ) |
177 |
175 176
|
eleqtrri |
|- 1 e. ( 0 ..^ ( # ` <" r s "> ) ) |
178 |
177
|
a1i |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> 1 e. ( 0 ..^ ( # ` <" r s "> ) ) ) |
179 |
119 152 153 122 178
|
splfv2a |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) = ( <" r s "> ` 1 ) ) |
180 |
|
s2fv1 |
|- ( s e. T -> ( <" r s "> ` 1 ) = s ) |
181 |
180
|
ad2antll |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( <" r s "> ` 1 ) = s ) |
182 |
179 181
|
eqtrd |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) = s ) |
183 |
182
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) = s ) |
184 |
183
|
difeq1d |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) = ( s \ _I ) ) |
185 |
184
|
dmeqd |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) = dom ( s \ _I ) ) |
186 |
170 185
|
eleqtrrd |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) ) |
187 |
|
fzosplitsni |
|- ( I e. ( ZZ>= ` 0 ) -> ( j e. ( 0 ..^ ( I + 1 ) ) <-> ( j e. ( 0 ..^ I ) \/ j = I ) ) ) |
188 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
189 |
187 188
|
eleq2s |
|- ( I e. NN0 -> ( j e. ( 0 ..^ ( I + 1 ) ) <-> ( j e. ( 0 ..^ I ) \/ j = I ) ) ) |
190 |
18 189
|
syl |
|- ( ph -> ( j e. ( 0 ..^ ( I + 1 ) ) <-> ( j e. ( 0 ..^ I ) \/ j = I ) ) ) |
191 |
190
|
adantr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( j e. ( 0 ..^ ( I + 1 ) ) <-> ( j e. ( 0 ..^ I ) \/ j = I ) ) ) |
192 |
|
fveq2 |
|- ( k = j -> ( W ` k ) = ( W ` j ) ) |
193 |
192
|
difeq1d |
|- ( k = j -> ( ( W ` k ) \ _I ) = ( ( W ` j ) \ _I ) ) |
194 |
193
|
dmeqd |
|- ( k = j -> dom ( ( W ` k ) \ _I ) = dom ( ( W ` j ) \ _I ) ) |
195 |
194
|
eleq2d |
|- ( k = j -> ( A e. dom ( ( W ` k ) \ _I ) <-> A e. dom ( ( W ` j ) \ _I ) ) ) |
196 |
195
|
notbid |
|- ( k = j -> ( -. A e. dom ( ( W ` k ) \ _I ) <-> -. A e. dom ( ( W ` j ) \ _I ) ) ) |
197 |
196
|
rspccva |
|- ( ( A. k e. ( 0 ..^ I ) -. A e. dom ( ( W ` k ) \ _I ) /\ j e. ( 0 ..^ I ) ) -> -. A e. dom ( ( W ` j ) \ _I ) ) |
198 |
9 197
|
sylan |
|- ( ( ph /\ j e. ( 0 ..^ I ) ) -> -. A e. dom ( ( W ` j ) \ _I ) ) |
199 |
198
|
adantlr |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> -. A e. dom ( ( W ` j ) \ _I ) ) |
200 |
4
|
ad2antrr |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> W e. Word T ) |
201 |
26
|
ad2antrr |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> I e. ( 0 ... ( I + 2 ) ) ) |
202 |
37
|
ad2antrr |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> ( I + 2 ) e. ( 0 ... ( # ` W ) ) ) |
203 |
122
|
adantr |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> <" r s "> e. Word T ) |
204 |
|
simpr |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> j e. ( 0 ..^ I ) ) |
205 |
200 201 202 203 204
|
splfv1 |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) = ( W ` j ) ) |
206 |
205
|
difeq1d |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) = ( ( W ` j ) \ _I ) ) |
207 |
206
|
dmeqd |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) = dom ( ( W ` j ) \ _I ) ) |
208 |
199 207
|
neleqtrrd |
|- ( ( ( ph /\ ( r e. T /\ s e. T ) ) /\ j e. ( 0 ..^ I ) ) -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) |
209 |
208
|
ex |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( j e. ( 0 ..^ I ) -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
210 |
209
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( j e. ( 0 ..^ I ) -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
211 |
|
simprr3 |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> -. A e. dom ( r \ _I ) ) |
212 |
|
0nn0 |
|- 0 e. NN0 |
213 |
|
2pos |
|- 0 < 2 |
214 |
|
elfzo0 |
|- ( 0 e. ( 0 ..^ 2 ) <-> ( 0 e. NN0 /\ 2 e. NN /\ 0 < 2 ) ) |
215 |
212 172 213 214
|
mpbir3an |
|- 0 e. ( 0 ..^ 2 ) |
216 |
215 176
|
eleqtrri |
|- 0 e. ( 0 ..^ ( # ` <" r s "> ) ) |
217 |
216
|
a1i |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> 0 e. ( 0 ..^ ( # ` <" r s "> ) ) ) |
218 |
119 152 153 122 217
|
splfv2a |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 0 ) ) = ( <" r s "> ` 0 ) ) |
219 |
32
|
addid1d |
|- ( ph -> ( I + 0 ) = I ) |
220 |
219
|
adantr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( I + 0 ) = I ) |
221 |
220
|
fveq2d |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 0 ) ) = ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) ) |
222 |
|
s2fv0 |
|- ( r e. T -> ( <" r s "> ` 0 ) = r ) |
223 |
222
|
ad2antrl |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( <" r s "> ` 0 ) = r ) |
224 |
218 221 223
|
3eqtr3d |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) = r ) |
225 |
224
|
difeq1d |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) = ( r \ _I ) ) |
226 |
225
|
dmeqd |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) = dom ( r \ _I ) ) |
227 |
226
|
eleq2d |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) <-> A e. dom ( r \ _I ) ) ) |
228 |
227
|
adantrr |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) <-> A e. dom ( r \ _I ) ) ) |
229 |
211 228
|
mtbird |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) ) |
230 |
|
fveq2 |
|- ( j = I -> ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) = ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) ) |
231 |
230
|
difeq1d |
|- ( j = I -> ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) = ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) ) |
232 |
231
|
dmeqd |
|- ( j = I -> dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) = dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) ) |
233 |
232
|
eleq2d |
|- ( j = I -> ( A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) <-> A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) ) ) |
234 |
233
|
notbid |
|- ( j = I -> ( -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) <-> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` I ) \ _I ) ) ) |
235 |
229 234
|
syl5ibrcom |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( j = I -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
236 |
210 235
|
jaod |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( ( j e. ( 0 ..^ I ) \/ j = I ) -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
237 |
191 236
|
sylbid |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( j e. ( 0 ..^ ( I + 1 ) ) -> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
238 |
237
|
ralrimiv |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) |
239 |
169 186 238
|
3jca |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
240 |
|
oveq2 |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( G gsum w ) = ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) ) |
241 |
240
|
eqeq1d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( G gsum w ) = ( _I |` D ) <-> ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( _I |` D ) ) ) |
242 |
|
fveqeq2 |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( # ` w ) = L <-> ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) ) |
243 |
241 242
|
anbi12d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) <-> ( ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( _I |` D ) /\ ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) ) ) |
244 |
|
fveq1 |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( w ` ( I + 1 ) ) = ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) ) |
245 |
244
|
difeq1d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( w ` ( I + 1 ) ) \ _I ) = ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) ) |
246 |
245
|
dmeqd |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> dom ( ( w ` ( I + 1 ) ) \ _I ) = dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) ) |
247 |
246
|
eleq2d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( A e. dom ( ( w ` ( I + 1 ) ) \ _I ) <-> A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) ) ) |
248 |
|
fveq1 |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( w ` j ) = ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) ) |
249 |
248
|
difeq1d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( w ` j ) \ _I ) = ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) |
250 |
249
|
dmeqd |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> dom ( ( w ` j ) \ _I ) = dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) |
251 |
250
|
eleq2d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( A e. dom ( ( w ` j ) \ _I ) <-> A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
252 |
251
|
notbid |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( -. A e. dom ( ( w ` j ) \ _I ) <-> -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
253 |
252
|
ralbidv |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) <-> A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) |
254 |
247 253
|
3anbi23d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) <-> ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) ) |
255 |
243 254
|
anbi12d |
|- ( w = ( W splice <. I , ( I + 2 ) , <" r s "> >. ) -> ( ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) <-> ( ( ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( _I |` D ) /\ ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) ) ) |
256 |
255
|
rspcev |
|- ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) e. Word T /\ ( ( ( G gsum ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = ( _I |` D ) /\ ( # ` ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( ( W splice <. I , ( I + 2 ) , <" r s "> >. ) ` j ) \ _I ) ) ) ) -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) |
257 |
125 168 239 256
|
syl12anc |
|- ( ( ph /\ ( ( r e. T /\ s e. T ) /\ ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) |
258 |
257
|
expr |
|- ( ( ph /\ ( r e. T /\ s e. T ) ) -> ( ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) ) |
259 |
258
|
rexlimdvva |
|- ( ph -> ( E. r e. T E. s e. T ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) ) |
260 |
2 3 88 90 8
|
psgnunilem1 |
|- ( ph -> ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( _I |` D ) \/ E. r e. T E. s e. T ( ( ( W ` I ) o. ( W ` ( I + 1 ) ) ) = ( r o. s ) /\ A e. dom ( s \ _I ) /\ -. A e. dom ( r \ _I ) ) ) ) |
261 |
118 259 260
|
mpjaod |
|- ( ph -> E. w e. Word T ( ( ( G gsum w ) = ( _I |` D ) /\ ( # ` w ) = L ) /\ ( ( I + 1 ) e. ( 0 ..^ L ) /\ A e. dom ( ( w ` ( I + 1 ) ) \ _I ) /\ A. j e. ( 0 ..^ ( I + 1 ) ) -. A e. dom ( ( w ` j ) \ _I ) ) ) ) |