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Theorem gsumspl 16228
Description: The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Hypotheses
Ref Expression
gsumspl.b  |-  B  =  ( Base `  M
)
gsumspl.m  |-  ( ph  ->  M  e.  Mnd )
gsumspl.s  |-  ( ph  ->  S  e. Word  B )
gsumspl.f  |-  ( ph  ->  F  e.  ( 0 ... T ) )
gsumspl.t  |-  ( ph  ->  T  e.  ( 0 ... ( # `  S
) ) )
gsumspl.x  |-  ( ph  ->  X  e. Word  B )
gsumspl.y  |-  ( ph  ->  Y  e. Word  B )
gsumspl.eq  |-  ( ph  ->  ( M  gsumg  X )  =  ( M  gsumg  Y ) )
Assertion
Ref Expression
gsumspl  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
gsumg  ( S splice  <. F ,  T ,  Y >. ) ) )

Proof of Theorem gsumspl
StepHypRef Expression
1 gsumspl.eq . . . 4  |-  ( ph  ->  ( M  gsumg  X )  =  ( M  gsumg  Y ) )
21oveq2d 6250 . . 3  |-  ( ph  ->  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M
) ( M  gsumg  X ) )  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
32oveq1d 6249 . 2  |-  ( ph  ->  ( ( ( M 
gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) ( +g  `  M ) ( M  gsumg  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( ( M 
gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) ( +g  `  M ) ( M  gsumg  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
4 gsumspl.s . . . . 5  |-  ( ph  ->  S  e. Word  B )
5 gsumspl.f . . . . 5  |-  ( ph  ->  F  e.  ( 0 ... T ) )
6 gsumspl.t . . . . 5  |-  ( ph  ->  T  e.  ( 0 ... ( # `  S
) ) )
7 gsumspl.x . . . . 5  |-  ( ph  ->  X  e. Word  B )
8 splval 12690 . . . . 5  |-  ( ( S  e. Word  B  /\  ( F  e.  (
0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) )  /\  X  e. Word  B ) )  -> 
( S splice  <. F ,  T ,  X >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  X ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) )
94, 5, 6, 7, 8syl13anc 1232 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  X >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  X ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) )
109oveq2d 6250 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) ++  X ) ++  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
11 gsumspl.m . . . 4  |-  ( ph  ->  M  e.  Mnd )
12 swrdcl 12607 . . . . . 6  |-  ( S  e. Word  B  ->  ( S substr  <. 0 ,  F >. )  e. Word  B )
134, 12syl 17 . . . . 5  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  B
)
14 ccatcl 12554 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  X  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) ++  X )  e. Word  B
)
1513, 7, 14syl2anc 659 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) ++  X )  e. Word  B )
16 swrdcl 12607 . . . . 5  |-  ( S  e. Word  B  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  B )
174, 16syl 17 . . . 4  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  B )
18 gsumspl.b . . . . 5  |-  B  =  ( Base `  M
)
19 eqid 2402 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
2018, 19gsumccat 16225 . . . 4  |-  ( ( M  e.  Mnd  /\  ( ( S substr  <. 0 ,  F >. ) ++  X )  e. Word  B  /\  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  B )  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) ++  X ) ++  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  X ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
2111, 15, 17, 20syl3anc 1230 . . 3  |-  ( ph  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) ++  X ) ++  ( S substr  <. T ,  ( # `  S ) >. )
) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  X ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
2218, 19gsumccat 16225 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( S substr  <. 0 ,  F >. )  e. Word  B  /\  X  e. Word  B )  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  X ) )  =  ( ( M 
gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) )
2311, 13, 7, 22syl3anc 1230 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  X ) )  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) )
2423oveq1d 6249 . . 3  |-  ( ph  ->  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  X ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) )  =  ( ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M
) ( M  gsumg  X ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
2510, 21, 243eqtrd 2447 . 2  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) ( +g  `  M ) ( M  gsumg  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
26 gsumspl.y . . . . 5  |-  ( ph  ->  Y  e. Word  B )
27 splval 12690 . . . . 5  |-  ( ( S  e. Word  B  /\  ( F  e.  (
0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) )  /\  Y  e. Word  B ) )  -> 
( S splice  <. F ,  T ,  Y >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  Y ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) )
284, 5, 6, 26, 27syl13anc 1232 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  Y >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  Y ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) )
2928oveq2d 6250 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  Y >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) ++  Y ) ++  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
30 ccatcl 12554 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  Y  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) ++  Y )  e. Word  B
)
3113, 26, 30syl2anc 659 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) ++  Y )  e. Word  B )
3218, 19gsumccat 16225 . . . 4  |-  ( ( M  e.  Mnd  /\  ( ( S substr  <. 0 ,  F >. ) ++  Y )  e. Word  B  /\  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  B )  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) ++  Y ) ++  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  Y ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
3311, 31, 17, 32syl3anc 1230 . . 3  |-  ( ph  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) ++  Y ) ++  ( S substr  <. T ,  ( # `  S ) >. )
) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  Y ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
3418, 19gsumccat 16225 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( S substr  <. 0 ,  F >. )  e. Word  B  /\  Y  e. Word  B )  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  Y ) )  =  ( ( M 
gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
3511, 13, 26, 34syl3anc 1230 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  Y ) )  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
3635oveq1d 6249 . . 3  |-  ( ph  ->  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) ++  Y ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) )  =  ( ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M
) ( M  gsumg  Y ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
3729, 33, 363eqtrd 2447 . 2  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  Y >. ) )  =  ( ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) ( +g  `  M ) ( M  gsumg  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
383, 25, 373eqtr4d 2453 1  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
gsumg  ( S splice  <. F ,  T ,  Y >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   <.cop 3977   <.cotp 3979   ` cfv 5525  (class class class)co 6234   0cc0 9442   ...cfz 11643   #chash 12359  Word cword 12490   ++ cconcat 12492   substr csubstr 12494   splice csplice 12495   Basecbs 14733   +g cplusg 14801    gsumg cgsu 14947   Mndcmnd 16135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-fzo 11768  df-seq 12062  df-hash 12360  df-word 12498  df-concat 12500  df-substr 12502  df-splice 12503  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-0g 14948  df-gsum 14949  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183
This theorem is referenced by:  psgnunilem2  16736
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