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Theorem psgnunilem2 27286
Description: Lemma for psgnuni 27290. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
psgnunilem2.g  |-  G  =  ( SymGrp `  D )
psgnunilem2.t  |-  T  =  ran  (pmTrsp `  D
)
psgnunilem2.d  |-  ( ph  ->  D  e.  V )
psgnunilem2.w  |-  ( ph  ->  W  e. Word  T )
psgnunilem2.id  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
psgnunilem2.l  |-  ( ph  ->  ( # `  W
)  =  L )
psgnunilem2.ix  |-  ( ph  ->  I  e.  ( 0..^ L ) )
psgnunilem2.a  |-  ( ph  ->  A  e.  dom  (
( W `  I
)  \  _I  )
)
psgnunilem2.al  |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )
)
psgnunilem2.in  |-  ( ph  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
Assertion
Ref Expression
psgnunilem2  |-  ( ph  ->  E. w  e. Word  T
( ( ( G 
gsumg  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  ) ) ) )
Distinct variable groups:    j, k, w, A    x, j, D, w    ph, j    j, G   
x, k, G, w   
j, I, k, w, x    T, j, w, x   
j, W, k, w, x    w, L, x
Allowed substitution hints:    ph( x, w, k)    A( x)    D( k)    T( k)    L( j, k)    V( x, w, j, k)

Proof of Theorem psgnunilem2
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnunilem2.w . . . . . . 7  |-  ( ph  ->  W  e. Word  T )
2 wrd0 11687 . . . . . . 7  |-  (/)  e. Word  T
3 splcl 11736 . . . . . . 7  |-  ( ( W  e. Word  T  /\  (/) 
e. Word  T )  ->  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T
)
41, 2, 3sylancl 644 . . . . . 6  |-  ( ph  ->  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T )
54adantr 452 . . . . 5  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( W splice  <.
I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T
)
6 fzossfz 11112 . . . . . . . . . . 11  |-  ( 0..^ L )  C_  (
0 ... L )
7 psgnunilem2.ix . . . . . . . . . . 11  |-  ( ph  ->  I  e.  ( 0..^ L ) )
86, 7sseldi 3306 . . . . . . . . . 10  |-  ( ph  ->  I  e.  ( 0 ... L ) )
9 elfznn0 11039 . . . . . . . . . 10  |-  ( I  e.  ( 0 ... L )  ->  I  e.  NN0 )
108, 9syl 16 . . . . . . . . 9  |-  ( ph  ->  I  e.  NN0 )
11 2nn0 10194 . . . . . . . . . 10  |-  2  e.  NN0
12 nn0addcl 10211 . . . . . . . . . 10  |-  ( ( I  e.  NN0  /\  2  e.  NN0 )  -> 
( I  +  2 )  e.  NN0 )
1310, 11, 12sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( I  +  2 )  e.  NN0 )
1410nn0red 10231 . . . . . . . . . 10  |-  ( ph  ->  I  e.  RR )
15 nn0addge1 10222 . . . . . . . . . 10  |-  ( ( I  e.  RR  /\  2  e.  NN0 )  ->  I  <_  ( I  + 
2 ) )
1614, 11, 15sylancl 644 . . . . . . . . 9  |-  ( ph  ->  I  <_  ( I  +  2 ) )
17 elfz2nn0 11038 . . . . . . . . 9  |-  ( I  e.  ( 0 ... ( I  +  2 ) )  <->  ( I  e.  NN0  /\  ( I  +  2 )  e. 
NN0  /\  I  <_  ( I  +  2 ) ) )
1810, 13, 16, 17syl3anbrc 1138 . . . . . . . 8  |-  ( ph  ->  I  e.  ( 0 ... ( I  + 
2 ) ) )
19 psgnunilem2.g . . . . . . . . . . 11  |-  G  =  ( SymGrp `  D )
20 psgnunilem2.t . . . . . . . . . . 11  |-  T  =  ran  (pmTrsp `  D
)
21 psgnunilem2.d . . . . . . . . . . 11  |-  ( ph  ->  D  e.  V )
22 psgnunilem2.id . . . . . . . . . . 11  |-  ( ph  ->  ( G  gsumg  W )  =  (  _I  |`  D )
)
23 psgnunilem2.l . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  =  L )
24 psgnunilem2.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  dom  (
( W `  I
)  \  _I  )
)
25 psgnunilem2.al . . . . . . . . . . 11  |-  ( ph  ->  A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )
)
2619, 20, 21, 1, 22, 23, 7, 24, 25psgnunilem5 27285 . . . . . . . . . 10  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ L ) )
27 fzofzp1 11144 . . . . . . . . . 10  |-  ( ( I  +  1 )  e.  ( 0..^ L )  ->  ( (
I  +  1 )  +  1 )  e.  ( 0 ... L
) )
2826, 27syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( I  + 
1 )  +  1 )  e.  ( 0 ... L ) )
2910nn0cnd 10232 . . . . . . . . . . 11  |-  ( ph  ->  I  e.  CC )
30 ax-1cn 9004 . . . . . . . . . . . 12  |-  1  e.  CC
3130a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  1  e.  CC )
3229, 31, 31addassd 9066 . . . . . . . . . 10  |-  ( ph  ->  ( ( I  + 
1 )  +  1 )  =  ( I  +  ( 1  +  1 ) ) )
33 df-2 10014 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
3433oveq2i 6051 . . . . . . . . . 10  |-  ( I  +  2 )  =  ( I  +  ( 1  +  1 ) )
3532, 34syl6reqr 2455 . . . . . . . . 9  |-  ( ph  ->  ( I  +  2 )  =  ( ( I  +  1 )  +  1 ) )
3623oveq2d 6056 . . . . . . . . 9  |-  ( ph  ->  ( 0 ... ( # `
 W ) )  =  ( 0 ... L ) )
3728, 35, 363eltr4d 2485 . . . . . . . 8  |-  ( ph  ->  ( I  +  2 )  e.  ( 0 ... ( # `  W
) ) )
382a1i 11 . . . . . . . 8  |-  ( ph  -> 
(/)  e. Word  T )
391, 18, 37, 38spllen 11738 . . . . . . 7  |-  ( ph  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( ( # `  W
)  +  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) ) ) )
40 hash0 11601 . . . . . . . . . . 11  |-  ( # `  (/) )  =  0
4140oveq1i 6050 . . . . . . . . . 10  |-  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) )  =  ( 0  -  ( ( I  +  2 )  -  I ) )
42 df-neg 9250 . . . . . . . . . 10  |-  -u (
( I  +  2 )  -  I )  =  ( 0  -  ( ( I  + 
2 )  -  I
) )
4341, 42eqtr4i 2427 . . . . . . . . 9  |-  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) )  =  -u ( ( I  + 
2 )  -  I
)
44 2cn 10026 . . . . . . . . . . 11  |-  2  e.  CC
45 pncan2 9268 . . . . . . . . . . 11  |-  ( ( I  e.  CC  /\  2  e.  CC )  ->  ( ( I  + 
2 )  -  I
)  =  2 )
4629, 44, 45sylancl 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( I  + 
2 )  -  I
)  =  2 )
4746negeqd 9256 . . . . . . . . 9  |-  ( ph  -> 
-u ( ( I  +  2 )  -  I )  =  -u
2 )
4843, 47syl5eq 2448 . . . . . . . 8  |-  ( ph  ->  ( ( # `  (/) )  -  ( ( I  + 
2 )  -  I
) )  =  -u
2 )
4923, 48oveq12d 6058 . . . . . . 7  |-  ( ph  ->  ( ( # `  W
)  +  ( (
# `  (/) )  -  ( ( I  + 
2 )  -  I
) ) )  =  ( L  +  -u
2 ) )
50 elfzel2 11013 . . . . . . . . . 10  |-  ( I  e.  ( 0 ... L )  ->  L  e.  ZZ )
518, 50syl 16 . . . . . . . . 9  |-  ( ph  ->  L  e.  ZZ )
5251zcnd 10332 . . . . . . . 8  |-  ( ph  ->  L  e.  CC )
53 negsub 9305 . . . . . . . 8  |-  ( ( L  e.  CC  /\  2  e.  CC )  ->  ( L  +  -u
2 )  =  ( L  -  2 ) )
5452, 44, 53sylancl 644 . . . . . . 7  |-  ( ph  ->  ( L  +  -u
2 )  =  ( L  -  2 ) )
5539, 49, 543eqtrd 2440 . . . . . 6  |-  ( ph  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 ) )
5655adantr 452 . . . . 5  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 ) )
57 splid 11737 . . . . . . . . 9  |-  ( ( 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 )
581, 18, 37, 57syl12anc 1182 . . . . . . . 8  |-  ( ph  ->  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. )  =  W )
5958oveq2d 6056 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. ) )  =  ( G  gsumg  W ) )
6059adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >. ) >. )
)  =  ( G 
gsumg  W ) )
61 eqid 2404 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
6219symggrp 15058 . . . . . . . . . 10  |-  ( D  e.  V  ->  G  e.  Grp )
6321, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
64 grpmnd 14772 . . . . . . . . 9  |-  ( G  e.  Grp  ->  G  e.  Mnd )
6563, 64syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Mnd )
6665adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  G  e.  Mnd )
6720, 19, 61symgtrf 27278 . . . . . . . . . 10  |-  T  C_  ( Base `  G )
68 sswrd 11692 . . . . . . . . . 10  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
6967, 68ax-mp 8 . . . . . . . . 9  |- Word  T  C_ Word  (
Base `  G )
7069, 1sseldi 3306 . . . . . . . 8  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
7170adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  W  e. Word  (
Base `  G )
)
7218adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  I  e.  ( 0 ... (
I  +  2 ) ) )
7337adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( I  +  2 )  e.  ( 0 ... ( # `
 W ) ) )
74 swrdcl 11721 . . . . . . . . 9  |-  ( W  e. Word  ( Base `  G
)  ->  ( W substr  <.
I ,  ( I  +  2 ) >.
)  e. Word  ( Base `  G ) )
7570, 74syl 16 . . . . . . . 8  |-  ( ph  ->  ( W substr  <. I ,  ( I  +  2 ) >. )  e. Word  ( Base `  G ) )
7675adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( W substr  <.
I ,  ( I  +  2 ) >.
)  e. Word  ( Base `  G ) )
77 wrd0 11687 . . . . . . . 8  |-  (/)  e. Word  ( Base `  G )
7877a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  (/)  e. Word  ( Base `  G ) )
7923oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 0..^ ( # `  W ) )  =  ( 0..^ L ) )
8026, 79eleqtrrd 2481 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  +  1 )  e.  ( 0..^ ( # `  W
) ) )
81 swrds2 11835 . . . . . . . . . . . 12  |-  ( ( W  e. Word  T  /\  I  e.  NN0  /\  (
I  +  1 )  e.  ( 0..^ (
# `  W )
) )  ->  ( W substr  <. I ,  ( I  +  2 )
>. )  =  <" ( W `  I
) ( W `  ( I  +  1
) ) "> )
821, 10, 80, 81syl3anc 1184 . . . . . . . . . . 11  |-  ( ph  ->  ( W substr  <. I ,  ( I  +  2 ) >. )  =  <" ( W `  I
) ( W `  ( I  +  1
) ) "> )
8382oveq2d 6056 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >. ) )  =  ( G  gsumg 
<" ( W `  I ) ( W `
 ( I  + 
1 ) ) "> ) )
84 wrdf 11688 . . . . . . . . . . . . . . 15  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
851, 84syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  W : ( 0..^ ( # `  W
) ) --> T )
8679feq2d 5540 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W : ( 0..^ ( # `  W
) ) --> T  <->  W :
( 0..^ L ) --> T ) )
8785, 86mpbid 202 . . . . . . . . . . . . 13  |-  ( ph  ->  W : ( 0..^ L ) --> T )
8887, 7ffvelrnd 5830 . . . . . . . . . . . 12  |-  ( ph  ->  ( W `  I
)  e.  T )
8967, 88sseldi 3306 . . . . . . . . . . 11  |-  ( ph  ->  ( W `  I
)  e.  ( Base `  G ) )
9087, 26ffvelrnd 5830 . . . . . . . . . . . 12  |-  ( ph  ->  ( W `  (
I  +  1 ) )  e.  T )
9167, 90sseldi 3306 . . . . . . . . . . 11  |-  ( ph  ->  ( W `  (
I  +  1 ) )  e.  ( Base `  G ) )
92 eqid 2404 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
9361, 92gsumws2 14743 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( W `  I )  e.  ( Base `  G
)  /\  ( W `  ( I  +  1 ) )  e.  (
Base `  G )
)  ->  ( G  gsumg  <" ( W `  I ) ( W `
 ( I  + 
1 ) ) "> )  =  ( ( W `  I
) ( +g  `  G
) ( W `  ( I  +  1
) ) ) )
9465, 89, 91, 93syl3anc 1184 . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg 
<" ( W `  I ) ( W `
 ( I  + 
1 ) ) "> )  =  ( ( W `  I
) ( +g  `  G
) ( W `  ( I  +  1
) ) ) )
9519, 61, 92symgov 15055 . . . . . . . . . . 11  |-  ( ( ( W `  I
)  e.  ( Base `  G )  /\  ( W `  ( I  +  1 ) )  e.  ( Base `  G
) )  ->  (
( W `  I
) ( +g  `  G
) ( W `  ( I  +  1
) ) )  =  ( ( W `  I )  o.  ( W `  ( I  +  1 ) ) ) )
9689, 91, 95syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( W `  I ) ( +g  `  G ) ( W `
 ( I  + 
1 ) ) )  =  ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) ) )
9783, 94, 963eqtrd 2440 . . . . . . . . 9  |-  ( ph  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >. ) )  =  ( ( W `  I )  o.  ( W `  ( I  +  1 ) ) ) )
9897adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 )
>. ) )  =  ( ( W `  I
)  o.  ( W `
 ( I  + 
1 ) ) ) )
99 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)
10019symgid 15059 . . . . . . . . . . 11  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
10121, 100syl 16 . . . . . . . . . 10  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
102 eqid 2404 . . . . . . . . . . 11  |-  ( 0g
`  G )  =  ( 0g `  G
)
103102gsum0 14735 . . . . . . . . . 10  |-  ( G 
gsumg  (/) )  =  ( 0g
`  G )
104101, 103syl6eqr 2454 . . . . . . . . 9  |-  ( ph  ->  (  _I  |`  D )  =  ( G  gsumg  (/) ) )
105104adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  (  _I  |`  D )  =  ( G  gsumg  (/) ) )
10698, 99, 1053eqtrd 2440 . . . . . . 7  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 )
>. ) )  =  ( G  gsumg  (/) ) )
10761, 66, 71, 72, 73, 76, 78, 106gsumspl 14744 . . . . . 6  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >. ) >. )
)  =  ( G 
gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
) )
10822adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  W )  =  (  _I  |`  D ) )
10960, 107, 1083eqtr3d 2444 . . . . 5  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  (  _I  |`  D ) )
110 fveq2 5687 . . . . . . . 8  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( # `  x
)  =  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
) )
111110eqeq1d 2412 . . . . . . 7  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( ( # `  x
)  =  ( L  -  2 )  <->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 ) ) )
112 oveq2 6048 . . . . . . . 8  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( G  gsumg  x )  =  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
) )
113112eqeq1d 2412 . . . . . . 7  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( ( G  gsumg  x )  =  (  _I  |`  D )  <-> 
( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  (  _I  |`  D ) ) )
114111, 113anbi12d 692 . . . . . 6  |-  ( x  =  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  ->  ( ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) )  <->  ( ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  ( L  -  2 )  /\  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  (  _I  |`  D ) ) ) )
115114rspcev 3012 . . . . 5  |-  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )  e. Word  T  /\  ( (
# `  ( W splice  <.
I ,  ( I  +  2 ) ,  (/) >. ) )  =  ( L  -  2 )  /\  ( G 
gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  (/) >. )
)  =  (  _I  |`  D ) ) )  ->  E. x  e. Word  T
( ( # `  x
)  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D )
) )
1165, 56, 109, 115syl12anc 1182 . . . 4  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
117 psgnunilem2.in . . . . 5  |-  ( ph  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
118117adantr 452 . . . 4  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  -.  E. x  e. Word  T ( ( # `  x )  =  ( L  -  2 )  /\  ( G  gsumg  x )  =  (  _I  |`  D ) ) )
119116, 118pm2.21dd 101 . . 3  |-  ( (
ph  /\  ( ( W `  I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )
)  ->  E. w  e. Word  T ( ( ( G  gsumg  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  ) ) ) )
120119ex 424 . 2  |-  ( ph  ->  ( ( ( W `
 I )  o.  ( W `  (
I  +  1 ) ) )  =  (  _I  |`  D )  ->  E. w  e. Word  T
( ( ( G 
gsumg  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  ) ) ) ) )
1211adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  W  e. Word  T )
122 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
r  e.  T )
123 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
s  e.  T )
124122, 123s2cld 11788 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  <" r s ">  e. Word  T )
125 splcl 11736 . . . . . . 7  |-  ( ( W  e. Word  T  /\  <" r s ">  e. Word  T )  ->  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T )
126121, 124, 125syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T )
127126adantrr 698 . . . . 5  |-  ( (
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 )
12865adantr 452 . . . . . . . 8  |-  ( (
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 )
12970adantr 452 . . . . . . . 8  |-  ( (
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 ) )
13018adantr 452 . . . . . . . 8  |-  ( (
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 ) ) )
13137adantr 452 . . . . . . . 8  |-  ( (
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 ) ) )
13269, 124sseldi 3306 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  <" r s ">  e. Word  ( Base `  G ) )
133132adantrr 698 . . . . . . . 8  |-  ( (
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 ) )
13475adantr 452 . . . . . . . 8  |-  ( (
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 ) )
135 simprr1 1005 . . . . . . . . 9  |-  ( (
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 ) )
13697adantr 452 . . . . . . . . 9  |-  ( (
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  gsumg  ( W substr  <. I ,  ( I  +  2 ) >.
) )  =  ( ( W `  I
)  o.  ( W `
 ( I  + 
1 ) ) ) )
13765adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  G  e.  Mnd )
13867a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  C_  ( Base `  G ) )
139138sselda 3308 . . . . . . . . . . . . 13  |-  ( (
ph  /\  r  e.  T )  ->  r  e.  ( Base `  G
) )
140139adantrr 698 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
r  e.  ( Base `  G ) )
141138sselda 3308 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  ( Base `  G
) )
142141adantrl 697 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
s  e.  ( Base `  G ) )
14361, 92gsumws2 14743 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  r  e.  ( Base `  G )  /\  s  e.  ( Base `  G
) )  ->  ( G  gsumg 
<" r s "> )  =  ( r ( +g  `  G
) s ) )
144137, 140, 142, 143syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( G  gsumg 
<" r s "> )  =  ( r ( +g  `  G
) s ) )
14519, 61, 92symgov 15055 . . . . . . . . . . . 12  |-  ( ( r  e.  ( Base `  G )  /\  s  e.  ( Base `  G
) )  ->  (
r ( +g  `  G
) s )  =  ( r  o.  s
) )
146140, 142, 145syl2anc 643 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( r ( +g  `  G ) s )  =  ( r  o.  s ) )
147144, 146eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( G  gsumg 
<" r s "> )  =  ( r  o.  s ) )
148147adantrr 698 . . . . . . . . 9  |-  ( (
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  gsumg  <" r
s "> )  =  ( r  o.  s ) )
149135, 136, 1483eqtr4rd 2447 . . . . . . . 8  |-  ( (
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  gsumg  <" r
s "> )  =  ( G  gsumg  ( W substr  <. I ,  ( I  +  2 ) >.
) ) )
15061, 128, 129, 130, 131, 133, 134, 149gsumspl 14744 . . . . . . 7  |-  ( (
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  gsumg  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >.
) >. ) ) )
15159adantr 452 . . . . . . 7  |-  ( (
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  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  ( W substr  <. I ,  ( I  +  2 ) >. ) >. )
)  =  ( G 
gsumg  W ) )
15222adantr 452 . . . . . . 7  |-  ( (
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  gsumg  W )  =  (  _I  |`  D ) )
153150, 151, 1523eqtrd 2440 . . . . . 6  |-  ( (
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  gsumg  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  (  _I  |`  D ) )
15418adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  I  e.  ( 0 ... ( I  + 
2 ) ) )
15537adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( I  +  2 )  e.  ( 0 ... ( # `  W
) ) )
156121, 154, 155, 124spllen 11738 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( # `  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  ( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) ) )
157 s2len 11806 . . . . . . . . . . . . 13  |-  ( # `  <" r s "> )  =  2
158157oveq1i 6050 . . . . . . . . . . . 12  |-  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) )  =  ( 2  -  (
( I  +  2 )  -  I ) )
15946oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 2  -  (
( I  +  2 )  -  I ) )  =  ( 2  -  2 ) )
16044subidi 9327 . . . . . . . . . . . . 13  |-  ( 2  -  2 )  =  0
161159, 160syl6eq 2452 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  -  (
( I  +  2 )  -  I ) )  =  0 )
162158, 161syl5eq 2448 . . . . . . . . . . 11  |-  ( ph  ->  ( ( # `  <" r s "> )  -  ( (
I  +  2 )  -  I ) )  =  0 )
163162oveq2d 6056 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) )  =  ( ( # `  W )  +  0 ) )
16423, 52eqeltrd 2478 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  W
)  e.  CC )
165164addid1d 9222 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  W
)  +  0 )  =  ( # `  W
) )
166163, 165, 233eqtrd 2440 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) )  =  L )
167166adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( # `  W
)  +  ( (
# `  <" r
s "> )  -  ( ( I  +  2 )  -  I ) ) )  =  L )
168156, 167eqtrd 2436 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( # `  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  L )
169168adantrr 698 . . . . . 6  |-  ( (
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 )
170153, 169jca 519 . . . . 5  |-  ( (
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 
gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  (  _I  |`  D )  /\  ( # `
 ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  L ) )
17126adantr 452 . . . . . 6  |-  ( (
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 ) )
172 simprr2 1006 . . . . . . 7  |-  ( (
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  ) )
173 1nn0 10193 . . . . . . . . . . . . . . 15  |-  1  e.  NN0
174 2nn 10089 . . . . . . . . . . . . . . 15  |-  2  e.  NN
175 1lt2 10098 . . . . . . . . . . . . . . 15  |-  1  <  2
176 elfzo0 11126 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ( 0..^ 2 )  <->  ( 1  e. 
NN0  /\  2  e.  NN  /\  1  <  2
) )
177173, 174, 175, 176mpbir3an 1136 . . . . . . . . . . . . . 14  |-  1  e.  ( 0..^ 2 )
178157oveq2i 6051 . . . . . . . . . . . . . 14  |-  ( 0..^ ( # `  <" r s "> ) )  =  ( 0..^ 2 )
179177, 178eleqtrri 2477 . . . . . . . . . . . . 13  |-  1  e.  ( 0..^ ( # `  <" r s "> ) )
180179a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
1  e.  ( 0..^ ( # `  <" r s "> ) ) )
181121, 154, 155, 124, 180splfv2a 11740 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  =  (
<" r s "> `  1 )
)
182 s2fv1 11805 . . . . . . . . . . . 12  |-  ( s  e.  T  ->  ( <" r s "> `  1 )  =  s )
183182ad2antll 710 . . . . . . . . . . 11  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( <" r s "> `  1
)  =  s )
184181, 183eqtrd 2436 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  =  s )
185184adantrr 698 . . . . . . . . 9  |-  ( (
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 )
186185difeq1d 3424 . . . . . . . 8  |-  ( (
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  ) )
187186dmeqd 5031 . . . . . . 7  |-  ( (
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  ) )
188172, 187eleqtrrd 2481 . . . . . 6  |-  ( (
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  )
)
189 fzosplitsni 11151 . . . . . . . . . . 11  |-  ( I  e.  ( ZZ>= `  0
)  ->  ( j  e.  ( 0..^ ( I  +  1 ) )  <-> 
( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
190 nn0uz 10476 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
191189, 190eleq2s 2496 . . . . . . . . . 10  |-  ( I  e.  NN0  ->  ( j  e.  ( 0..^ ( I  +  1 ) )  <->  ( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
19210, 191syl 16 . . . . . . . . 9  |-  ( ph  ->  ( j  e.  ( 0..^ ( I  + 
1 ) )  <->  ( j  e.  ( 0..^ I )  \/  j  =  I ) ) )
193192adantr 452 . . . . . . . 8  |-  ( (
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 ) ) )
194 fveq2 5687 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  j  ->  ( W `  k )  =  ( W `  j ) )
195194difeq1d 3424 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  j  ->  (
( W `  k
)  \  _I  )  =  ( ( W `
 j )  \  _I  ) )
196195dmeqd 5031 . . . . . . . . . . . . . . . . 17  |-  ( k  =  j  ->  dom  ( ( W `  k )  \  _I  )  =  dom  ( ( W `  j ) 
\  _I  ) )
197196eleq2d 2471 . . . . . . . . . . . . . . . 16  |-  ( k  =  j  ->  ( A  e.  dom  ( ( W `  k ) 
\  _I  )  <->  A  e.  dom  ( ( W `  j )  \  _I  ) ) )
198197notbid 286 . . . . . . . . . . . . . . 15  |-  ( k  =  j  ->  ( -.  A  e.  dom  ( ( W `  k )  \  _I  ) 
<->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) ) )
199198rspccva 3011 . . . . . . . . . . . . . 14  |-  ( ( A. k  e.  ( 0..^ I )  -.  A  e.  dom  (
( W `  k
)  \  _I  )  /\  j  e.  (
0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) )
20025, 199sylan 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) )
201200adantlr 696 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( W `  j )  \  _I  ) )
2021ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  W  e. Word  T
)
20318ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  I  e.  ( 0 ... ( I  +  2 ) ) )
20437ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  ( I  + 
2 )  e.  ( 0 ... ( # `  W ) ) )
205124adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  <" r s ">  e. Word  T
)
206 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  j  e.  ( 0..^ I ) )
207202, 203, 204, 205, 206splfv1 11739 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  =  ( W `
 j ) )
208207difeq1d 3424 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  j )  \  _I  )  =  ( ( W `  j )  \  _I  ) )
209208dmeqd 5031 . . . . . . . . . . . 12  |-  ( ( ( 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  ) )
210201, 209neleqtrrd 2500 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  T  /\  s  e.  T )
)  /\  j  e.  ( 0..^ I ) )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
)
211210ex 424 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( j  e.  ( 0..^ I )  ->  -.  A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
) )
212211adantrr 698 . . . . . . . . 9  |-  ( (
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  )
) )
213 simprr3 1007 . . . . . . . . . . 11  |-  ( (
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  ) )
214 0nn0 10192 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  NN0
215 2pos 10038 . . . . . . . . . . . . . . . . . . . 20  |-  0  <  2
216 elfzo0 11126 . . . . . . . . . . . . . . . . . . . 20  |-  ( 0  e.  ( 0..^ 2 )  <->  ( 0  e. 
NN0  /\  2  e.  NN  /\  0  <  2
) )
217214, 174, 215, 216mpbir3an 1136 . . . . . . . . . . . . . . . . . . 19  |-  0  e.  ( 0..^ 2 )
218217, 178eleqtrri 2477 . . . . . . . . . . . . . . . . . 18  |-  0  e.  ( 0..^ ( # `  <" r s "> ) )
219218a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
0  e.  ( 0..^ ( # `  <" r s "> ) ) )
220121, 154, 155, 124, 219splfv2a 11740 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  0 ) )  =  (
<" r s "> `  0 )
)
22129addid1d 9222 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( I  +  0 )  =  I )
222221adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( I  +  0 )  =  I )
223222fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  0 ) )  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I ) )
224 s2fv0 11804 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  T  ->  ( <" r s "> `  0 )  =  r )
225224ad2antrl 709 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( <" r s "> `  0
)  =  r )
226220, 223, 2253eqtr3d 2444 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I )  =  r )
227226difeq1d 3424 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )  =  ( r  \  _I  ) )
228227dmeqd 5031 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  ->  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )  =  dom  ( r  \  _I  ) )
229228eleq2d 2471 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( r  e.  T  /\  s  e.  T ) )  -> 
( A  e.  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  I
)  \  _I  )  <->  A  e.  dom  ( r 
\  _I  ) ) )
230229adantrr 698 . . . . . . . . . . 11  |-  ( (
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  )
) )
231213, 230mtbird 293 . . . . . . . . . 10  |-  ( (
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  )
)
232 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( j  =  I  ->  (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I ) )
233232difeq1d 3424 . . . . . . . . . . . . 13  |-  ( j  =  I  ->  (
( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  )  =  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  I )  \  _I  ) )
234233dmeqd 5031 . . . . . . . . . . . 12  |-  ( j  =  I  ->  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )  =  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  I )  \  _I  ) )
235234eleq2d 2471 . . . . . . . . . . 11  |-  ( 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  ) ) )
236235notbid 286 . . . . . . . . . 10  |-  ( 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  ) ) )
237231, 236syl5ibrcom 214 . . . . . . . . 9  |-  ( (
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  ) ) )
238212, 237jaod 370 . . . . . . . 8  |-  ( (
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  ) ) )
239193, 238sylbid 207 . . . . . . 7  |-  ( (
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  )
) )
240239ralrimiv 2748 . . . . . 6  |-  ( (
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  ) )
241171, 188, 2403jca 1134 . . . . 5  |-  ( (
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  ) ) )
242 oveq2 6048 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( G  gsumg  w )  =  ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
) )
243242eqeq1d 2412 . . . . . . . 8  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( G  gsumg  w )  =  (  _I  |`  D )  <-> 
( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
)  =  (  _I  |`  D ) ) )
244 fveq2 5687 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( # `  w
)  =  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )
) )
245244eqeq1d 2412 . . . . . . . 8  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( # `  w
)  =  L  <->  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. )
)  =  L ) )
246243, 245anbi12d 692 . . . . . . 7  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( ( G 
gsumg  w )  =  (  _I  |`  D )  /\  ( # `  w
)  =  L )  <-> 
( ( G  gsumg  ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) )  =  (  _I  |`  D )  /\  ( # `  ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. )
)  =  L ) ) )
247 fveq1 5686 . . . . . . . . . . 11  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( w `  (
I  +  1 ) )  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  ( I  +  1 ) ) )
248247difeq1d 3424 . . . . . . . . . 10  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( w `  ( I  +  1
) )  \  _I  )  =  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  ( I  +  1 ) )  \  _I  ) )
249248dmeqd 5031 . . . . . . . . 9  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  dom  ( ( w `
 ( I  + 
1 ) )  \  _I  )  =  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  (
I  +  1 ) )  \  _I  )
)
250249eleq2d 2471 . . . . . . . 8  |-  ( 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  ) ) )
251 fveq1 5686 . . . . . . . . . . . . 13  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( w `  j
)  =  ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r s "> >. ) `  j ) )
252251difeq1d 3424 . . . . . . . . . . . 12  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( w `  j )  \  _I  )  =  ( (
( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. ) `  j )  \  _I  ) )
253252dmeqd 5031 . . . . . . . . . . 11  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  dom  ( ( w `
 j )  \  _I  )  =  dom  ( ( ( W splice  <. I ,  ( I  +  2 ) , 
<" r s "> >. ) `  j
)  \  _I  )
)
254253eleq2d 2471 . . . . . . . . . 10  |-  ( 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  ) ) )
255254notbid 286 . . . . . . . . 9  |-  ( 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  )
) )
256255ralbidv 2686 . . . . . . . 8  |-  ( 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  ) ) )
257250, 2563anbi23d 1257 . . . . . . 7  |-  ( 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  )
) ) )
258246, 257anbi12d 692 . . . . . 6  |-  ( w  =  ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  ->  ( ( ( ( G  gsumg  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  gsumg  ( 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  ) ) ) ) )
259258rspcev 3012 . . . . 5  |-  ( ( ( W splice  <. I ,  ( I  +  2 ) ,  <" r
s "> >. )  e. Word  T  /\  ( ( ( G  gsumg  ( 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 
gsumg  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  ) ) ) )
260127, 170, 241, 259syl12anc 1182 . . . 4  |-  ( (
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 
gsumg  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  ) ) ) )
261260expr 599 . . 3  |-  ( (
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  gsumg  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  ) ) ) ) )
262261rexlimdvva 2797 . 2  |-  ( 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  gsumg  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  ) ) ) ) )
26320, 21, 88, 90, 24psgnunilem1 27284 . 2  |-  ( 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  ) ) ) )
264120, 262, 263mpjaod 371 1  |-  ( ph  ->  E. w  e. Word  T
( ( ( G 
gsumg  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  ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    \ cdif 3277    C_ wss 3280   (/)c0 3588   <.cop 3777   <.cotp 3778   class class class wbr 4172    _I cid 4453   dom cdm 4837   ran crn 4838    |` cres 4839    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   substr csubstr 11675   splice csplice 11676   <"cs2 11760   Basecbs 13424   +g cplusg 13484   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639   Grpcgrp 14640   SymGrpcsymg 15047  pmTrspcpmtr 27252
This theorem is referenced by:  psgnunilem3  27287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-tset 13503  df-0g 13682  df-gsum 13683  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-subg 14896  df-symg 15048  df-pmtr 27253
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