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Theorem ccatpfx 11155
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 10238 . . . . . . . 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 11133 . . . . . . 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 9495 . . . . . . 7 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> Y e. ZZ )
7 elfzelz 10149 . . . . . . . . 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 11106 . . . . . . 7 |- ( ( S e. Word A /\ Y e. ZZ /\ Z e. ZZ ) -> ( S substr <. Y , Z >. ) e. Word A )
115, 6, 9, 10syl3anc 1250 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( S substr <. Y , Z >. ) e. Word A )
12 ccatcl 11052 . . . . . 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 11011 . . . . 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 11054 . . . . . . . 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 10186 . . . . . . . . . . 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 11139 . . . . . . . . 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 11108 . . . . . . . . 9 |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
24233expb 1207 . . . . . . . 8 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( S substr <. Y , Z >. ) ) = ( Z - Y ) )
2522, 24oveq12d 5964 . . . . . . 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 10149 . . . . . . . . . 10 |- ( Y e. ( 0 ... Z ) -> Y e. ZZ )
2726zcnd 9498 . . . . . . . . 9 |- ( Y e. ( 0 ... Z ) -> Y e. CC )
287zcnd 9498 . . . . . . . . 9 |- ( Z e. ( 0 ... (♯ ` S ) ) -> Z e. CC )
29 pncan3 8282 . . . . . . . . 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 2242 . . . . . 6 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) = Z )
3332oveq2d 5962 . . . . 5 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) ) ) = ( 0..^ Z ) )
3433fneq2d 5366 . . . 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 11137 . . . 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 10302 . . . . . 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 5962 . . . . . . . . . 10 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( 0..^ (♯ ` ( S prefix Y ) ) ) = ( 0..^ Y ) )
4544eleq2d 2275 . . . . . . . . 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 11056 . . . . . . . 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 1250 . . . . . . 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 11138 . . . . . . . 8 |- ( ( S e. Word A /\ Y e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Y ) ) -> ( ( S prefix Y ) ` x ) = ( S ` x ) )
525, 49, 50, 51syl2an3an 1311 . . . . . . 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 2238 . . . . . 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 2238 . . . . . . . . . . 11 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( (♯ ` ( S prefix Y ) ) + (♯ ` ( S substr <. Y , Z >. ) ) ) = Z )
5722, 56oveq12d 5964 . . . . . . . . . 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 2275 . . . . . . . . 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 11057 . . . . . . . 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 1250 . . . . . . 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 1207 . . . . . . . 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 5962 . . . . . . . . . 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 10325 . . . . . . . . . . 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 8373 . . . . . . . . . . . . . 14 |- ( Y e. ( 0 ... Z ) -> ( Y - Y ) = 0 )
7069oveq1d 5961 . . . . . . . . . . . . 13 |- ( Y e. ( 0 ... Z ) -> ( ( Y - Y )..^ ( Z - Y ) ) = ( 0..^ ( Z - Y ) ) )
7170eleq2d 2275 . . . . . . . . . . . 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 2282 . . . . . . . 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 11109 . . . . . . . 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 595 . . . . . . 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 5961 . . . . . . . . . 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 10271 . . . . . . . . . . 11 |- ( x e. ( Y..^ Z ) -> x e. ZZ )
8180zcnd 9498 . . . . . . . . . 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 8283 . . . . . . . . . 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 2238 . . . . . . . 8 |- ( ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) /\ x e. ( Y..^ Z ) ) -> ( ( x - (♯ ` ( S prefix Y ) ) ) + Y ) = x )
8685fveq2d 5582 . . . . . . 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 2242 . . . . . 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 799 . . . . 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 11138 . . . . . 6 |- ( ( S e. Word A /\ Z e. ( 0 ... (♯ ` S ) ) /\ x e. ( 0..^ Z ) ) -> ( ( S prefix Z ) ` x ) = ( S ` x ) )
91903expa 1206 . . . . 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 2241 . . 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 5682 . 2 |- ( ( S e. Word A /\ ( Y e. ( 0 ... Z ) /\ Z e. ( 0 ... (♯ ` S ) ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) )
95943impb 1202 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 710   /\ w3a 981   = wceq 1373   e. wcel 2176   <.cop 3636   Fn wfn 5267   ` cfv 5272  (class class class)co 5946   CCcc 7925   0cc0 7927   + caddc 7930   - cmin 8245   NN0cn0 9297   ZZcz 9374   ...cfz 10132  ..^cfzo 10266  ♯chash 10922  Word cword 10996   ++ cconcat 11049   substr csubstr 11101   prefix cpfx 11128
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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-fzo 10267  df-ihash 10923  df-word 10997  df-concat 11050  df-substr 11102  df-pfx 11129
This theorem is referenced by: (None)
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