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

Step	Hyp	Ref	Expression
1		gsumspl.eq	. . . 4 |- ( ph -> ( M gsumg X ) = ( M gsumg Y ) )
2	1	oveq2d 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 ) ) )
3	2	oveq1d 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 ) >. ) ) )
9	4, 5, 6, 7, 8	syl13anc 1232	. . . 4 |- ( ph -> ( S splice <. F , T , X >. ) = ( ( ( S substr <. 0 , F >. ) ++ X ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
10	9	oveq2d 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 )
13	4, 12	syl 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 )
15	13, 7, 14	syl2anc 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 )
17	4, 16	syl 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 )
20	18, 19	gsumccat 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 ) >. ) ) ) )
21	11, 15, 17, 20	syl3anc 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 ) >. ) ) ) )
22	18, 19	gsumccat 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 ) ) )
23	11, 13, 7, 22	syl3anc 1230	. . . 4 |- ( ph -> ( M gsumg ( ( S substr <. 0 , F >. ) ++ X ) ) = ( ( M gsumg ( S substr <. 0 , F >. ) ) ( +g ` M ) ( M gsumg X ) ) )
24	23	oveq1d 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 ) >. ) ) ) )
25	10, 21, 24	3eqtrd 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 ) >. ) ) )
28	4, 5, 6, 26, 27	syl13anc 1232	. . . 4 |- ( ph -> ( S splice <. F , T , Y >. ) = ( ( ( S substr <. 0 , F >. ) ++ Y ) ++ ( S substr <. T , ( # ` S ) >. ) ) )
29	28	oveq2d 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 )
31	13, 26, 30	syl2anc 659	. . . 4 |- ( ph -> ( ( S substr <. 0 , F >. ) ++ Y ) e. Word B )
32	18, 19	gsumccat 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 ) >. ) ) ) )
33	11, 31, 17, 32	syl3anc 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 ) >. ) ) ) )
34	18, 19	gsumccat 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 ) ) )
35	11, 13, 26, 34	syl3anc 1230	. . . 4 |- ( ph -> ( M gsumg ( ( S substr <. 0 , F >. ) ++ Y ) ) = ( ( M gsumg ( S substr <. 0 , F >. ) ) ( +g ` M ) ( M gsumg Y ) ) )
36	35	oveq1d 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 ) >. ) ) ) )
37	29, 33, 36	3eqtrd 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 ) >. ) ) ) )
38	3, 25, 37	3eqtr4d 2453	1 |- ( ph -> ( M gsumg ( S splice <. F , T , X >. ) ) = ( M gsumg ( S splice <. F , T , Y >. ) ) )

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   gsum_g 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