Theorem splvalpfx 41435

Description: Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)

Assertion

Ref	Expression
splvalpfx	|- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( S splice <. F , T , R >. ) = ( ( ( S prefix F ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )

Proof of Theorem splvalpfx

Step	Hyp	Ref	Expression
1		splval 13502	. 2 |- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( S splice <. F , T , R >. ) = ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
2		pfxval 41383	. . . . . 6 |- ( ( S e. V /\ F e. NN0 ) -> ( S prefix F ) = ( S substr <. 0 , F >. ) )
3	2	3ad2antr1 1226	. . . . 5 |- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( S prefix F ) = ( S substr <. 0 , F >. ) )
4	3	eqcomd 2628	. . . 4 |- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( S substr <. 0 , F >. ) = ( S prefix F ) )
5	4	oveq1d 6665	. . 3 |- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( ( S substr <. 0 , F >. ) ++ R ) = ( ( S prefix F ) ++ R ) )
6	5	oveq1d 6665	. 2 |- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( ( ( S substr <. 0 , F >. ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) = ( ( ( S prefix F ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
7	1, 6	eqtrd 2656	1 |- ( ( S e. V /\ ( F e. NN0 /\ T e. X /\ R e. Y ) ) -> ( S splice <. F , T , R >. ) = ( ( ( S prefix F ) ++ R ) ++ ( S substr <. T , ( # ` S ) >. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   ` cfv 5888  (class class class)co 6650   0cc0 9936   NN0cn0 11292   #chash 13117   ++ cconcat 13293   substr csubstr 13295   splice csplice 13296   prefix cpfx 41381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-splice 13304  df-pfx 41382

This theorem is referenced by: (None)