Theorem gsumspl 15828

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 6291	. . 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 6290	. 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 12677	. . . . 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 ) >. ) ) )
9	4, 5, 6, 7, 8	syl13anc 1225	. . . 4 |- ( ph -> ( S splice <. F , T , X >. ) = ( ( ( S substr <. 0 , F >. ) concat X ) concat ( S substr <. T , ( # ` S ) >. ) ) )
10	9	oveq2d 6291	. . 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 12596	. . . . . 6 |- ( S e. Word B -> ( S substr <. 0 , F >. ) e. Word B )
13	4, 12	syl 16	. . . . 5 |- ( ph -> ( S substr <. 0 , F >. ) e. Word B )
14		ccatcl 12545	. . . . 5 |- ( ( ( S substr <. 0 , F >. ) e. Word B /\ X e. Word B ) -> ( ( S substr <. 0 , F >. ) concat X ) e. Word B )
15	13, 7, 14	syl2anc 661	. . . 4 |- ( ph -> ( ( S substr <. 0 , F >. ) concat X ) e. Word B )
16		swrdcl 12596	. . . . 5 |- ( S e. Word B -> ( S substr <. T , ( # ` S ) >. ) e. Word B )
17	4, 16	syl 16	. . . 4 |- ( ph -> ( S substr <. T , ( # ` S ) >. ) e. Word B )
18		gsumspl.b	. . . . 5 |- B = ( Base ` M )
19		eqid 2460	. . . . 5 |- ( +g ` M ) = ( +g ` M )
20	18, 19	gsumccat 15825	. . . 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 ) >. ) ) ) )
21	11, 15, 17, 20	syl3anc 1223	. . 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 ) >. ) ) ) )
22	18, 19	gsumccat 15825	. . . . 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 ) ) )
23	11, 13, 7, 22	syl3anc 1223	. . . 4 |- ( ph -> ( M gsumg ( ( S substr <. 0 , F >. ) concat X ) ) = ( ( M gsumg ( S substr <. 0 , F >. ) ) ( +g ` M ) ( M gsumg X ) ) )
24	23	oveq1d 6290	. . 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 ) >. ) ) ) )
25	10, 21, 24	3eqtrd 2505	. 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 12677	. . . . 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 ) >. ) ) )
28	4, 5, 6, 26, 27	syl13anc 1225	. . . 4 |- ( ph -> ( S splice <. F , T , Y >. ) = ( ( ( S substr <. 0 , F >. ) concat Y ) concat ( S substr <. T , ( # ` S ) >. ) ) )
29	28	oveq2d 6291	. . 3 |- ( ph -> ( M gsumg ( S splice <. F , T , Y >. ) ) = ( M gsumg ( ( ( S substr <. 0 , F >. ) concat Y ) concat ( S substr <. T , ( # ` S ) >. ) ) ) )
30		ccatcl 12545	. . . . 5 |- ( ( ( S substr <. 0 , F >. ) e. Word B /\ Y e. Word B ) -> ( ( S substr <. 0 , F >. ) concat Y ) e. Word B )
31	13, 26, 30	syl2anc 661	. . . 4 |- ( ph -> ( ( S substr <. 0 , F >. ) concat Y ) e. Word B )
32	18, 19	gsumccat 15825	. . . 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 ) >. ) ) ) )
33	11, 31, 17, 32	syl3anc 1223	. . 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 ) >. ) ) ) )
34	18, 19	gsumccat 15825	. . . . 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 ) ) )
35	11, 13, 26, 34	syl3anc 1223	. . . 4 |- ( ph -> ( M gsumg ( ( S substr <. 0 , F >. ) concat Y ) ) = ( ( M gsumg ( S substr <. 0 , F >. ) ) ( +g ` M ) ( M gsumg Y ) ) )
36	35	oveq1d 6290	. . 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 ) >. ) ) ) )
37	29, 33, 36	3eqtrd 2505	. 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 2511	1 |- ( ph -> ( M gsumg ( S splice <. F , T , X >. ) ) = ( M gsumg ( S splice <. F , T , Y >. ) ) )

Syntax hints:   -> wi 4   = wceq 1374   e. wcel 1762   <.cop 4026   <.cotp 4028   ` cfv 5579  (class class class)co 6275   0cc0 9481   ...cfz 11661   #chash 12360  Word cword 12487   concat cconcat 12489   substr csubstr 12491   splice csplice 12492   Basecbs 14479   +g cplusg 14544   gsum_g cgsu 14685   Mndcmnd 15715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-word 12495  df-concat 12497  df-substr 12499  df-splice 12500  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-gsum 14687  df-mnd 15721  df-submnd 15771

This theorem is referenced by:  psgnunilem2  16309