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

Step	Hyp	Ref	Expression
1		gsumspl.eq	. . . 4 |- ( ph -> ( M gsumg X ) = ( M gsumg Y ) )
2	1	oveq2d 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 ) ) )
3	2	oveq1d 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 ) >. ) ) )
9	4, 5, 6, 7, 8	syl13anc 1221	. . . 4 |- ( ph -> ( S splice <. F , T , X >. ) = ( ( ( S substr <. 0 , F >. ) concat X ) concat ( S substr <. T , ( # ` S ) >. ) ) )
10	9	oveq2d 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 )
13	4, 12	syl 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 )
15	13, 7, 14	syl2anc 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 )
17	4, 16	syl 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 )
20	18, 19	gsumccat 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 ) >. ) ) ) )
21	11, 15, 17, 20	syl3anc 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 ) >. ) ) ) )
22	18, 19	gsumccat 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 ) ) )
23	11, 13, 7, 22	syl3anc 1219	. . . 4 |- ( ph -> ( M gsumg ( ( S substr <. 0 , F >. ) concat X ) ) = ( ( M gsumg ( S substr <. 0 , F >. ) ) ( +g ` M ) ( M gsumg X ) ) )
24	23	oveq1d 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 ) >. ) ) ) )
25	10, 21, 24	3eqtrd 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 ) >. ) ) )
28	4, 5, 6, 26, 27	syl13anc 1221	. . . 4 |- ( ph -> ( S splice <. F , T , Y >. ) = ( ( ( S substr <. 0 , F >. ) concat Y ) concat ( S substr <. T , ( # ` S ) >. ) ) )
29	28	oveq2d 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 )
31	13, 26, 30	syl2anc 661	. . . 4 |- ( ph -> ( ( S substr <. 0 , F >. ) concat Y ) e. Word B )
32	18, 19	gsumccat 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 ) >. ) ) ) )
33	11, 31, 17, 32	syl3anc 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 ) >. ) ) ) )
34	18, 19	gsumccat 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 ) ) )
35	11, 13, 26, 34	syl3anc 1219	. . . 4 |- ( ph -> ( M gsumg ( ( S substr <. 0 , F >. ) concat Y ) ) = ( ( M gsumg ( S substr <. 0 , F >. ) ) ( +g ` M ) ( M gsumg Y ) ) )
36	35	oveq1d 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 ) >. ) ) ) )
37	29, 33, 36	3eqtrd 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 ) >. ) ) ) )
38	3, 25, 37	3eqtr4d 2505	1 |- ( ph -> ( M gsumg ( S splice <. F , T , X >. ) ) = ( M gsumg ( S splice <. F , T , Y >. ) ) )

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   gsum_g 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