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Theorem gsumspl 14777
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 6088 . . 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 6087 . 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 11768 . . . . 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 1186 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  X >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  X ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
109oveq2d 6088 . . 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 11754 . . . . . 6  |-  ( S  e. Word  B  ->  ( S substr  <. 0 ,  F >. )  e. Word  B )
134, 12syl 16 . . . . 5  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  B
)
14 ccatcl 11731 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  X  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B
)
1513, 7, 14syl2anc 643 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B )
16 swrdcl 11754 . . . . 5  |-  ( S  e. Word  B  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  B )
174, 16syl 16 . . . 4  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  B )
18 gsumspl.b . . . . 5  |-  B  =  ( Base `  M
)
19 eqid 2435 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
2018, 19gsumccat 14775 . . . 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 1184 . . 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 14775 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  X )
)  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) )
2423oveq1d 6087 . . 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 2471 . 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 11768 . . . . 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 1186 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  Y >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
2928oveq2d 6088 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  Y >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
30 ccatcl 11731 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  Y  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B
)
3113, 26, 30syl2anc 643 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B )
3218, 19gsumccat 14775 . . . 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 1184 . . 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 14775 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  Y )
)  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
3635oveq1d 6087 . . 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 2471 . 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 2477 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 1652    e. wcel 1725   <.cop 3809   <.cotp 3810   ` cfv 5445  (class class class)co 6072   0cc0 8979   ...cfz 11032   #chash 11606  Word cword 11705   concat cconcat 11706   substr csubstr 11708   splice csplice 11709   Basecbs 13457   +g cplusg 13517    gsumg cgsu 13712   Mndcmnd 14672
This theorem is referenced by:  psgnunilem2  27333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-fzo 11124  df-seq 11312  df-hash 11607  df-word 11711  df-concat 11712  df-substr 11714  df-splice 11715  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-0g 13715  df-gsum 13716  df-mnd 14678  df-submnd 14727
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