MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumspl Structured version   Unicode version

Theorem gsumspl 15642
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 6217 . . 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 6216 . 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 12512 . . . . 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 1221 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  X >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  X ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
109oveq2d 6217 . . 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 12434 . . . . . 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 12393 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  X  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B
)
1513, 7, 14syl2anc 661 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  X )  e. Word  B )
16 swrdcl 12434 . . . . 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 2454 . . . . 5  |-  ( +g  `  M )  =  ( +g  `  M )
2018, 19gsumccat 15639 . . . 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 1219 . . 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 15639 . . . . 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 1219 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  X )
)  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  X ) ) )
2423oveq1d 6216 . . 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 2499 . 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 12512 . . . . 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 1221 . . . 4  |-  ( ph  ->  ( S splice  <. F ,  T ,  Y >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
2928oveq2d 6217 . . 3  |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  Y >. ) )  =  ( M 
gsumg  ( ( ( S substr  <. 0 ,  F >. ) concat  Y ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
30 ccatcl 12393 . . . . 5  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  B  /\  Y  e. Word  B )  ->  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B
)
3113, 26, 30syl2anc 661 . . . 4  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  Y )  e. Word  B )
3218, 19gsumccat 15639 . . . 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 1219 . . 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 15639 . . . . 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 1219 . . . 4  |-  ( ph  ->  ( M  gsumg  ( ( S substr  <. 0 ,  F >. ) concat  Y )
)  =  ( ( M  gsumg  ( S substr  <. 0 ,  F >. ) ) ( +g  `  M ) ( M  gsumg  Y ) ) )
3635oveq1d 6216 . . 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 2499 . 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 2505 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 1370    e. wcel 1758   <.cop 3992   <.cotp 3994   ` cfv 5527  (class class class)co 6201   0cc0 9394   ...cfz 11555   #chash 12221  Word cword 12340   concat cconcat 12342   substr csubstr 12344   splice csplice 12345   Basecbs 14293   +g cplusg 14358    gsumg cgsu 14499   Mndcmnd 15529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-ot 3995  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925  df-hash 12222  df-word 12348  df-concat 12350  df-substr 12352  df-splice 12353  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-gsum 14501  df-mnd 15535  df-submnd 15585
This theorem is referenced by:  psgnunilem2  16121
  Copyright terms: Public domain W3C validator