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Theorem gsumspl 14515
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 5916 . . 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 5915 . 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 11513 . . . . 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 5916 . . 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 11499 . . . . . 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 11476 . . . . 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 11499 . . . . 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 2316 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
2018, 19gsumccat 14513 . . . 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 14513 . . . . 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 5915 . . 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 2352 . 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 11513 . . . . 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 5916 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  Y >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
30 ccatcl 11476 . . . . 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 14513 . . . 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 14513 . . . . 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 5915 . . 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 2352 . 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 2358 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 1633    e. wcel 1701   <.cop 3677   <.cotp 3678   ` cfv 5292  (class class class)co 5900   0cc0 8782   ...cfz 10829   #chash 11384  Word cword 11450   concat cconcat 11451   substr csubstr 11453   splice csplice 11454   Basecbs 13195   +g cplusg 13255    gsumg cgsu 13450   Mndcmnd 14410
This theorem is referenced by:  psgnunilem2  26566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-ot 3684  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-fzo 10918  df-seq 11094  df-hash 11385  df-word 11456  df-concat 11457  df-substr 11459  df-splice 11460  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-0g 13453  df-gsum 13454  df-mnd 14416  df-submnd 14465
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