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Theorem ccatpfx 11393
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 10448 . . . . . . . 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 11370 . . . . . . 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 9698 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
7 elfzelz 10359 . . . . . . . . 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 11342 . . . . . . 7 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
115, 6, 9, 10syl3anc 1274 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
12 ccatcl 11281 . . . . . 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 11239 . . . . 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 11283 . . . . . . . 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 10396 . . . . . . . . . . 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 11377 . . . . . . . . 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 11344 . . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
24233expb 1231 . . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
2522, 24oveq12d 6068 . . . . . . 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 10359 . . . . . . . . . 10 |- ( Y e. ( 0 ... Z ) -> Y e. ZZ )
2726zcnd 9701 . . . . . . . . 9 |- ( Y e. ( 0 ... Z ) -> Y e. CC )
287zcnd 9701 . . . . . . . . 9 |- ( Z e. ( 0 ... (♯ ` S ) ) -> Z e. CC )
29 pncan3 8481 . . . . . . . . 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 2269 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = Z )
3332oveq2d 6066 . . . . 5 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) = ( 0..^ Z ) )
3433fneq2d 5447 . . . 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 11375 . . . 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 10512 . . . . . 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 6066 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( S prefix Y ) ) ) = ( 0..^ Y ) )
4544eleq2d 2302 . . . . . . . . 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 11285 . . . . . . . 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 1274 . . . . . . 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 11376 . . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Y ) ) -> ( ( S prefix Y ) ` x ) = ( S ` x ) )
525, 49, 50, 51syl2an3an 1335 . . . . . . 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 2265 . . . . . 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 2265 . . . . . . . . . . 11 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = Z )
5722, 56oveq12d 6068 . . . . . . . . . 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 2302 . . . . . . . . 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 11286 . . . . . . . 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 1274 . . . . . . 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 1231 . . . . . . . 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 6066 . . . . . . . . . 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 10539 . . . . . . . . . . 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 8572 . . . . . . . . . . . . . 14 |- ( Y e. ( 0 ... Z ) -> ( Y - Y ) = 0 )
7069oveq1d 6065 . . . . . . . . . . . . 13 |- ( Y e. ( 0 ... Z ) -> ( ( Y - Y )..^ ( Z - Y ) ) = ( 0..^ ( Z - Y ) ) )
7170eleq2d 2302 . . . . . . . . . . . 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 2309 . . . . . . . 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 11345 . . . . . . . 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 6065 . . . . . . . . . 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 10481 . . . . . . . . . . 11 |- ( x e. ( Y..^ Z ) -> x e. ZZ )
8180zcnd 9701 . . . . . . . . . 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 8482 . . . . . . . . . 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 2265 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = x )
8685fveq2d 5674 . . . . . . 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 2269 . . . . . 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 805 . . . . 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 11376 . . . . . 6 |- ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
91903expa 1230 . . . . 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 2268 . . 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 5778 . 2 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )
95943impb 1226 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2203   <.cop 3692   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   + caddc 8130   - cmin 8444   NN0cn0 9496   ZZcz 9577   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337   prefix cpfx 11364
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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-substr 11338  df-pfx 11365
This theorem is referenced by:  pfxcctswrd  11402  wrdeqs1cat  11412
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