Theorem ofcccat 30620

Description: Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 5-Oct-2018.)

Hypotheses

Ref	Expression
ofcccat.1	|- ( ph -> F e. Word S )
ofcccat.2	|- ( ph -> G e. Word S )
ofcccat.3	|- ( ph -> K e. T )

Assertion

Ref	Expression
ofcccat	|- ( ph -> ( ( F ++ G )∘𝑓/𝑐 R K ) = ( ( F∘𝑓/𝑐 R K ) ++ ( G∘𝑓/𝑐 R K ) ) )

Proof of Theorem ofcccat

Step	Hyp	Ref	Expression
1		ofcccat.1	. . 3 |- ( ph -> F e. Word S )
2		ofcccat.2	. . 3 |- ( ph -> G e. Word S )
3		ofcccat.3	. . . 4 |- ( ph -> K e. T )
4		fconst6g 6094	. . . 4 |- ( K e. T -> ( ( 0..^ ( # ` F ) ) X. { K } ) : ( 0..^ ( # ` F ) ) --> T )
5		iswrdi 13309	. . . 4 |- ( ( ( 0..^ ( # ` F ) ) X. { K } ) : ( 0..^ ( # ` F ) ) --> T -> ( ( 0..^ ( # ` F ) ) X. { K } ) e. Word T )
6	3, 4, 5	3syl 18	. . 3 |- ( ph -> ( ( 0..^ ( # ` F ) ) X. { K } ) e. Word T )
7		fconst6g 6094	. . . 4 |- ( K e. T -> ( ( 0..^ ( # ` G ) ) X. { K } ) : ( 0..^ ( # ` G ) ) --> T )
8		iswrdi 13309	. . . 4 |- ( ( ( 0..^ ( # ` G ) ) X. { K } ) : ( 0..^ ( # ` G ) ) --> T -> ( ( 0..^ ( # ` G ) ) X. { K } ) e. Word T )
9	3, 7, 8	3syl 18	. . 3 |- ( ph -> ( ( 0..^ ( # ` G ) ) X. { K } ) e. Word T )
10		fzofi 12773	. . . . 5 |- ( 0..^ ( # ` F ) ) e. Fin
11		snfi 8038	. . . . 5 |- { K } e. Fin
12		hashxp 13221	. . . . 5 |- ( ( ( 0..^ ( # ` F ) ) e. Fin /\ { K } e. Fin ) -> ( # ` ( ( 0..^ ( # ` F ) ) X. { K } ) ) = ( ( # ` ( 0..^ ( # ` F ) ) ) x. ( # ` { K } ) ) )
13	10, 11, 12	mp2an 708	. . . 4 |- ( # ` ( ( 0..^ ( # ` F ) ) X. { K } ) ) = ( ( # ` ( 0..^ ( # ` F ) ) ) x. ( # ` { K } ) )
14		wrdfin 13323	. . . . . . . 8 |- ( F e. Word S -> F e. Fin )
15		hashcl 13147	. . . . . . . 8 |- ( F e. Fin -> ( # ` F ) e. NN0 )
16	1, 14, 15	3syl 18	. . . . . . 7 |- ( ph -> ( # ` F ) e. NN0 )
17		hashfzo0 13217	. . . . . . 7 |- ( ( # ` F ) e. NN0 -> ( # ` ( 0..^ ( # ` F ) ) ) = ( # ` F ) )
18	16, 17	syl 17	. . . . . 6 |- ( ph -> ( # ` ( 0..^ ( # ` F ) ) ) = ( # ` F ) )
19		hashsng 13159	. . . . . . 7 |- ( K e. T -> ( # ` { K } ) = 1 )
20	3, 19	syl 17	. . . . . 6 |- ( ph -> ( # ` { K } ) = 1 )
21	18, 20	oveq12d 6668	. . . . 5 |- ( ph -> ( ( # ` ( 0..^ ( # ` F ) ) ) x. ( # ` { K } ) ) = ( ( # ` F ) x. 1 ) )
22	16	nn0cnd 11353	. . . . . 6 |- ( ph -> ( # ` F ) e. CC )
23	22	mulid1d 10057	. . . . 5 |- ( ph -> ( ( # ` F ) x. 1 ) = ( # ` F ) )
24	21, 23	eqtrd 2656	. . . 4 |- ( ph -> ( ( # ` ( 0..^ ( # ` F ) ) ) x. ( # ` { K } ) ) = ( # ` F ) )
25	13, 24	syl5req 2669	. . 3 |- ( ph -> ( # ` F ) = ( # ` ( ( 0..^ ( # ` F ) ) X. { K } ) ) )
26		fzofi 12773	. . . . 5 |- ( 0..^ ( # ` G ) ) e. Fin
27		hashxp 13221	. . . . 5 |- ( ( ( 0..^ ( # ` G ) ) e. Fin /\ { K } e. Fin ) -> ( # ` ( ( 0..^ ( # ` G ) ) X. { K } ) ) = ( ( # ` ( 0..^ ( # ` G ) ) ) x. ( # ` { K } ) ) )
28	26, 11, 27	mp2an 708	. . . 4 |- ( # ` ( ( 0..^ ( # ` G ) ) X. { K } ) ) = ( ( # ` ( 0..^ ( # ` G ) ) ) x. ( # ` { K } ) )
29		wrdfin 13323	. . . . . . . 8 |- ( G e. Word S -> G e. Fin )
30		hashcl 13147	. . . . . . . 8 |- ( G e. Fin -> ( # ` G ) e. NN0 )
31	2, 29, 30	3syl 18	. . . . . . 7 |- ( ph -> ( # ` G ) e. NN0 )
32		hashfzo0 13217	. . . . . . 7 |- ( ( # ` G ) e. NN0 -> ( # ` ( 0..^ ( # ` G ) ) ) = ( # ` G ) )
33	31, 32	syl 17	. . . . . 6 |- ( ph -> ( # ` ( 0..^ ( # ` G ) ) ) = ( # ` G ) )
34	33, 20	oveq12d 6668	. . . . 5 |- ( ph -> ( ( # ` ( 0..^ ( # ` G ) ) ) x. ( # ` { K } ) ) = ( ( # ` G ) x. 1 ) )
35	31	nn0cnd 11353	. . . . . 6 |- ( ph -> ( # ` G ) e. CC )
36	35	mulid1d 10057	. . . . 5 |- ( ph -> ( ( # ` G ) x. 1 ) = ( # ` G ) )
37	34, 36	eqtrd 2656	. . . 4 |- ( ph -> ( ( # ` ( 0..^ ( # ` G ) ) ) x. ( # ` { K } ) ) = ( # ` G ) )
38	28, 37	syl5req 2669	. . 3 |- ( ph -> ( # ` G ) = ( # ` ( ( 0..^ ( # ` G ) ) X. { K } ) ) )
39	1, 2, 6, 9, 25, 38	ofccat 13708	. 2 |- ( ph -> ( ( F ++ G ) oF R ( ( ( 0..^ ( # ` F ) ) X. { K } ) ++ ( ( 0..^ ( # ` G ) ) X. { K } ) ) ) = ( ( F oF R ( ( 0..^ ( # ` F ) ) X. { K } ) ) ++ ( G oF R ( ( 0..^ ( # ` G ) ) X. { K } ) ) ) )
40		ccatcl 13359	. . . . . 6 |- ( ( F e. Word S /\ G e. Word S ) -> ( F ++ G ) e. Word S )
41	1, 2, 40	syl2anc 693	. . . . 5 |- ( ph -> ( F ++ G ) e. Word S )
42		wrdf 13310	. . . . 5 |- ( ( F ++ G ) e. Word S -> ( F ++ G ) : ( 0..^ ( # ` ( F ++ G ) ) ) --> S )
43	41, 42	syl 17	. . . 4 |- ( ph -> ( F ++ G ) : ( 0..^ ( # ` ( F ++ G ) ) ) --> S )
44		ovexd 6680	. . . 4 |- ( ph -> ( 0..^ ( # ` ( F ++ G ) ) ) e. _V )
45	43, 44, 3	ofcof 30169	. . 3 |- ( ph -> ( ( F ++ G )∘𝑓/𝑐 R K ) = ( ( F ++ G ) oF R ( ( 0..^ ( # ` ( F ++ G ) ) ) X. { K } ) ) )
46		ccatlen 13360	. . . . . . . 8 |- ( ( F e. Word S /\ G e. Word S ) -> ( # ` ( F ++ G ) ) = ( ( # ` F ) + ( # ` G ) ) )
47	1, 2, 46	syl2anc 693	. . . . . . 7 |- ( ph -> ( # ` ( F ++ G ) ) = ( ( # ` F ) + ( # ` G ) ) )
48	47	oveq2d 6666	. . . . . 6 |- ( ph -> ( 0..^ ( # ` ( F ++ G ) ) ) = ( 0..^ ( ( # ` F ) + ( # ` G ) ) ) )
49	48	xpeq1d 5138	. . . . 5 |- ( ph -> ( ( 0..^ ( # ` ( F ++ G ) ) ) X. { K } ) = ( ( 0..^ ( ( # ` F ) + ( # ` G ) ) ) X. { K } ) )
50		eqid 2622	. . . . . 6 |- ( ( 0..^ ( # ` F ) ) X. { K } ) = ( ( 0..^ ( # ` F ) ) X. { K } )
51		eqid 2622	. . . . . 6 |- ( ( 0..^ ( # ` G ) ) X. { K } ) = ( ( 0..^ ( # ` G ) ) X. { K } )
52		eqid 2622	. . . . . 6 |- ( ( 0..^ ( ( # ` F ) + ( # ` G ) ) ) X. { K } ) = ( ( 0..^ ( ( # ` F ) + ( # ` G ) ) ) X. { K } )
53	50, 51, 52, 3, 16, 31	ccatmulgnn0dir 30619	. . . . 5 |- ( ph -> ( ( ( 0..^ ( # ` F ) ) X. { K } ) ++ ( ( 0..^ ( # ` G ) ) X. { K } ) ) = ( ( 0..^ ( ( # ` F ) + ( # ` G ) ) ) X. { K } ) )
54	49, 53	eqtr4d 2659	. . . 4 |- ( ph -> ( ( 0..^ ( # ` ( F ++ G ) ) ) X. { K } ) = ( ( ( 0..^ ( # ` F ) ) X. { K } ) ++ ( ( 0..^ ( # ` G ) ) X. { K } ) ) )
55	54	oveq2d 6666	. . 3 |- ( ph -> ( ( F ++ G ) oF R ( ( 0..^ ( # ` ( F ++ G ) ) ) X. { K } ) ) = ( ( F ++ G ) oF R ( ( ( 0..^ ( # ` F ) ) X. { K } ) ++ ( ( 0..^ ( # ` G ) ) X. { K } ) ) ) )
56	45, 55	eqtrd 2656	. 2 |- ( ph -> ( ( F ++ G )∘𝑓/𝑐 R K ) = ( ( F ++ G ) oF R ( ( ( 0..^ ( # ` F ) ) X. { K } ) ++ ( ( 0..^ ( # ` G ) ) X. { K } ) ) ) )
57		wrdf 13310	. . . . 5 |- ( F e. Word S -> F : ( 0..^ ( # ` F ) ) --> S )
58	1, 57	syl 17	. . . 4 |- ( ph -> F : ( 0..^ ( # ` F ) ) --> S )
59	10	a1i 11	. . . 4 |- ( ph -> ( 0..^ ( # ` F ) ) e. Fin )
60	58, 59, 3	ofcof 30169	. . 3 |- ( ph -> ( F∘𝑓/𝑐 R K ) = ( F oF R ( ( 0..^ ( # ` F ) ) X. { K } ) ) )
61		wrdf 13310	. . . . 5 |- ( G e. Word S -> G : ( 0..^ ( # ` G ) ) --> S )
62	2, 61	syl 17	. . . 4 |- ( ph -> G : ( 0..^ ( # ` G ) ) --> S )
63	26	a1i 11	. . . 4 |- ( ph -> ( 0..^ ( # ` G ) ) e. Fin )
64	62, 63, 3	ofcof 30169	. . 3 |- ( ph -> ( G∘𝑓/𝑐 R K ) = ( G oF R ( ( 0..^ ( # ` G ) ) X. { K } ) ) )
65	60, 64	oveq12d 6668	. 2 |- ( ph -> ( ( F∘𝑓/𝑐 R K ) ++ ( G∘𝑓/𝑐 R K ) ) = ( ( F oF R ( ( 0..^ ( # ` F ) ) X. { K } ) ) ++ ( G oF R ( ( 0..^ ( # ` G ) ) X. { K } ) ) ) )
66	39, 56, 65	3eqtr4d 2666	1 |- ( ph -> ( ( F ++ G )∘𝑓/𝑐 R K ) = ( ( F∘𝑓/𝑐 R K ) ++ ( G∘𝑓/𝑐 R K ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293  ∘_𝑓/𝑐cofc 30157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-ofc 30158

This theorem is referenced by:  ofcs2  30622