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Theorem ccatpfx 11281
Description: Concatenating a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)
Assertion
Ref Expression
ccatpfx |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )
Proof of Theorem ccatpfx
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elfznn0 10348 . . . . . . . 8 |- ( Y e. ( 0 ... Z ) -> Y e. NN0 )
21ad2antrl 490 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. NN0 )
3 pfxclg 11258 . . . . . . 7 |- ( ( S e. Word A /\ Y e. NN0 ) -> ( S prefix Y ) e. Word A )
42, 3syldan 282 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S prefix Y ) e. Word A )
5 simpl 109 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> S e. Word A )
62nn0zd 9599 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
7 elfzelz 10259 . . . . . . . . 9 |- ( Z e. ( 0 ... (♯ ` S ) ) -> Z e. ZZ )
87adantl 277 . . . . . . . 8 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Z e. ZZ )
98adantl 277 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Z e. ZZ )
10 swrdclg 11230 . . . . . . 7 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
115, 6, 9, 10syl3anc 1273 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
12 ccatcl 11169 . . . . . 6 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) e. Word A )
134, 11, 12syl2anc 411 . . . . 5 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) e. Word A )
14 wrdfn 11127 . . . . 5 |- ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) e. Word A -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) )
1513, 14syl 14 . . . 4 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) )
16 ccatlen 11171 . . . . . . . 8 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) )
174, 11, 16syl2anc 411 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) )
18 fzass4 10296 . . . . . . . . . . 11 |- ( ( Y e. ( 0 ... (♯ ` S ) ) /\ Z e. ( Y ... (♯ ` S ) ) ) <-> ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
1918biimpri 133 . . . . . . . . . 10 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( Y e. ( 0 ... (♯ ` S ) ) /\ Z e. ( Y ... (♯ ` S ) ) ) )
2019simpld 112 . . . . . . . . 9 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
21 pfxlen 11265 . . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S prefix Y ) ) = Y )
2220, 21sylan2 286 . . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S prefix Y ) ) = Y )
23 swrdlen 11232 . . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
24233expb 1230 . . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
2522, 24oveq12d 6035 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = ( Y + ( Z - Y ) ) )
26 elfzelz 10259 . . . . . . . . . 10 |- ( Y e. ( 0 ... Z ) -> Y e. ZZ )
2726zcnd 9602 . . . . . . . . 9 |- ( Y e. ( 0 ... Z ) -> Y e. CC )
287zcnd 9602 . . . . . . . . 9 |- ( Z e. ( 0 ... (♯ ` S ) ) -> Z e. CC )
29 pncan3 8386 . . . . . . . . 9 |- ( ( Y e. CC /\ Z e. CC ) -> ( Y + ( Z - Y ) ) = Z )
3027, 28, 29syl2an 289 . . . . . . . 8 |- ( ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( Y + ( Z - Y ) ) = Z )
3130adantl 277 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( Y + ( Z - Y ) ) = Z )
3217, 25, 313eqtrd 2268 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = Z )
3332oveq2d 6033 . . . . 5 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) = ( 0..^ Z ) )
3433fneq2d 5421 . . . 4 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) <-> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ Z ) ) )
3515, 34mpbid 147 . . 3 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) Fn ( 0..^ Z ) )
36 pfxfn 11263 . . . 4 |- ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( S prefix Z ) Fn ( 0..^ Z ) )
3736adantrl 478 . . 3 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S prefix Z ) Fn ( 0..^ Z ) )
38 id 19 . . . . . 6 |- ( x e. ( 0..^ Z ) -> x e. ( 0..^ Z ) )
3926ad2antrl 490 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
40 fzospliti 10412 . . . . . 6 |- ( ( x e. ( 0..^ Z ) /\ Y e. ZZ ) -> ( x e. ( 0..^ Y ) \/ x e. ( Y..^ Z ) ) )
4138, 39, 40syl2anr 290 . . . . 5 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( x e. ( 0..^ Y ) \/ x e. ( Y..^ Z ) ) )
424adantr 276 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( S prefix Y ) e. Word A )
4311adantr 276 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( S substr <. Y , Z >. ) e. Word A )
4422oveq2d 6033 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( S prefix Y ) ) ) = ( 0..^ Y ) )
4544eleq2d 2301 . . . . . . . . 9 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( 0..^ (♯ ` ( S prefix Y ) ) ) <-> x e. ( 0..^ Y ) ) )
4645biimpar 297 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> x e. ( 0..^ (♯ ` ( S prefix Y ) ) ) )
47 ccatval1 11173 . . . . . . . 8 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A /\ x e. ( 0..^ (♯ ` ( S prefix Y ) ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S prefix Y ) ` x ) )
4842, 43, 46, 47syl3anc 1273 . . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S prefix Y ) ` x ) )
4920adantl 277 . . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ( 0 ... (♯ ` S ) ) )
50 id 19 . . . . . . . 8 |- ( x e. ( 0..^ Y ) -> x e. ( 0..^ Y ) )
51 pfxfv 11264 . . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Y ) ) -> ( ( S prefix Y ) ` x ) = ( S ` x ) )
525, 49, 50, 51syl2an3an 1334 . . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( ( S prefix Y ) ` x ) = ( S ` x ) )
5348, 52eqtrd 2264 . . . . . 6 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Y ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
544adantr 276 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( S prefix Y ) e. Word A )
5511adantr 276 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( S substr <. Y , Z >. ) e. Word A )
5625, 31eqtrd 2264 . . . . . . . . . . 11 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = Z )
5722, 56oveq12d 6035 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) = ( Y..^ Z ) )
5857eleq2d 2301 . . . . . . . . 9 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x e. ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) <-> x e. ( Y..^ Z ) ) )
5958biimpar 297 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> x e. ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) )
60 ccatval2 11174 . . . . . . . 8 |- ( ( ( S prefix Y ) e. Word A /\ ( S substr <. Y , Z >. ) e. Word A /\ x e. ( (♯ ` ( S prefix Y ) )..^ ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) )
6154, 55, 59, 60syl3anc 1273 . . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) )
62 id 19 . . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
63623expb 1230 . . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) )
6422oveq2d 6033 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( x - (♯ ` ( S prefix Y ) ) ) = ( x - Y ) )
6564adantr 276 . . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - (♯ ` ( S prefix Y ) ) ) = ( x - Y ) )
66 id 19 . . . . . . . . . . 11 |- ( x e. ( Y..^ Z ) -> x e. ( Y..^ Z ) )
67 fzosubel 10438 . . . . . . . . . . 11 |- ( ( x e. ( Y..^ Z ) /\ Y e. ZZ ) -> ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) )
6866, 39, 67syl2anr 290 . . . . . . . . . 10 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) )
6927subidd 8477 . . . . . . . . . . . . . 14 |- ( Y e. ( 0 ... Z ) -> ( Y - Y ) = 0 )
7069oveq1d 6032 . . . . . . . . . . . . 13 |- ( Y e. ( 0 ... Z ) -> ( ( Y - Y )..^ ( Z - Y ) ) = ( 0..^ ( Z - Y ) ) )
7170eleq2d 2301 . . . . . . . . . . . 12 |- ( Y e. ( 0 ... Z ) -> ( ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) <-> ( x - Y ) e. ( 0..^ ( Z - Y ) ) ) )
7271ad2antrl 490 . . . . . . . . . . 11 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) <-> ( x - Y ) e. ( 0..^ ( Z - Y ) ) ) )
7372adantr 276 . . . . . . . . . 10 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - Y ) e. ( ( Y - Y )..^ ( Z - Y ) ) <-> ( x - Y ) e. ( 0..^ ( Z - Y ) ) ) )
7468, 73mpbid 147 . . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - Y ) e. ( 0..^ ( Z - Y ) ) )
7565, 74eqeltrd 2308 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( x - (♯ ` ( S prefix Y ) ) ) e. ( 0..^ ( Z - Y ) ) )
76 swrdfv 11233 . . . . . . . 8 |- ( ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) /\ ( x - (♯ ` ( S prefix Y ) ) ) e. ( 0..^ ( Z - Y ) ) ) -> ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) = ( S ` ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) ) )
7763, 75, 76syl2an2r 599 . . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( S substr <. Y , Z >. ) ` ( x - (♯ ` ( S prefix Y ) ) ) ) = ( S ` ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) ) )
7864oveq1d 6032 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = ( ( x - Y ) + Y ) )
7978adantr 276 . . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = ( ( x - Y ) + Y ) )
80 elfzoelz 10381 . . . . . . . . . . 11 |- ( x e. ( Y..^ Z ) -> x e. ZZ )
8180zcnd 9602 . . . . . . . . . 10 |- ( x e. ( Y..^ Z ) -> x e. CC )
8227ad2antrl 490 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. CC )
83 npcan 8387 . . . . . . . . . 10 |- ( ( x e. CC /\ Y e. CC ) -> ( ( x - Y ) + Y ) = x )
8481, 82, 83syl2anr 290 . . . . . . . . 9 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - Y ) + Y ) = x )
8579, 84eqtrd 2264 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = x )
8685fveq2d 5643 . . . . . . 7 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( S ` ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) ) = ( S ` x ) )
8761, 77, 863eqtrd 2268 . . . . . 6 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
8853, 87jaodan 804 . . . . 5 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ ( x e. ( 0..^ Y ) \/ x e. ( Y..^ Z ) ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
8941, 88syldan 282 . . . 4 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( S ` x ) )
90 pfxfv 11264 . . . . . 6 |- ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
91903expa 1229 . . . . 5 |- ( ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
9291adantlrl 482 . . . 4 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
9389, 92eqtr4d 2267 . . 3 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( 0..^ Z ) ) -> ( ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ` x ) = ( ( S prefix Z ) ` x ) )
9435, 37, 93eqfnfvd 5747 . 2 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )
95943impb 1225 1 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   = wceq 1397   e. wcel 2202   <.cop 3672   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   + caddc 8034   - cmin 8349   NN0cn0 9401   ZZcz 9478   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Word cword 11112   ++ cconcat 11166   substr csubstr 11225   prefix cpfx 11252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-concat 11167  df-substr 11226  df-pfx 11253
This theorem is referenced by:  pfxcctswrd  11290  wrdeqs1cat  11300
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