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Theorem gsumspl 14466
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 5874 . . 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 5873 . 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 11466 . . . . 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 >. ) concat  X ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
94, 5, 6, 7, 8syl13anc 1184 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  X >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  X ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
109oveq2d 5874 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  X ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
11 gsumspl.m . . . 4  |-  ( ph  ->  M  e.  Mnd )
12 swrdcl 11452 . . . . . 6  |-  ( S  e. Word  B  ->  ( S substr  <. 0 ,  F >. )  e. Word  B )
134, 12syl 15 . . . . 5  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  B
)
14 ccatcl 11429 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  X  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B
)
1513, 7, 14syl2anc 642 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B )
16 swrdcl 11452 . . . . 5  |-  ( S  e. Word  B  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  B )
174, 16syl 15 . . . 4  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  B )
18 gsumspl.b . . . . 5  |-  B  =  ( Base `  M
)
19 eqid 2283 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
2018, 19gsumccat 14464 . . . 4  |-  ( ( M  e.  Mnd  /\  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B  /\  ( S substr  <. T ,  ( # `  S ) >. )  e. Word  B )  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  X ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  X ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
2111, 15, 17, 20syl3anc 1182 . . 3  |-  ( ph  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  X ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  X ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
2218, 19gsumccat 14464 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( S substr  <. 0 ,  F >. )  e. Word  B  /\  X  e. Word  B )  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  X ) )  =  ( ( M 
gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) )
2311, 13, 7, 22syl3anc 1182 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  X )
)  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) )
2423oveq1d 5873 . . 3  |-  ( ph  ->  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  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 2319 . 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 11466 . . . . 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 >. ) concat  Y ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
284, 5, 6, 26, 27syl13anc 1184 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  Y >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
2928oveq2d 5874 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  Y >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
30 ccatcl 11429 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  Y  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B
)
3113, 26, 30syl2anc 642 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B )
3218, 19gsumccat 14464 . . . 4  |-  ( ( M  e.  Mnd  /\  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B  /\  ( S substr  <. T ,  ( # `  S ) >. )  e. Word  B )  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  Y ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
3311, 31, 17, 32syl3anc 1182 . . 3  |-  ( ph  ->  ( M  gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  Y ) ) ( +g  `  M
) ( M  gsumg  ( S substr  <. T ,  ( # `  S ) >. )
) ) )
3418, 19gsumccat 14464 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( S substr  <. 0 ,  F >. )  e. Word  B  /\  Y  e. Word  B )  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  Y ) )  =  ( ( M 
gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
3511, 13, 26, 34syl3anc 1182 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  Y )
)  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
3635oveq1d 5873 . . 3  |-  ( ph  ->  ( ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  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 2319 . 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 2325 1  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
gsumg  ( S splice  <. F ,  T ,  Y >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   <.cop 3643   <.cotp 3644   ` cfv 5255  (class class class)co 5858   0cc0 8737   ...cfz 10782   #chash 11337  Word cword 11403   concat cconcat 11404   substr csubstr 11406   splice csplice 11407   Basecbs 13148   +g cplusg 13208    gsumg cgsu 13401   Mndcmnd 14361
This theorem is referenced by:  psgnunilem2  27418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-substr 11412  df-splice 11413  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416
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