Theorem ccatmulgnn0dir 28122

Description: Concatenation of words follow the rule mulgnn0dir 15958 (although applying mulgnn0dir 15958 would require S to be a set). In this case A is <" K "> to the power M in the free monoid. (Contributed by Thierry Arnoux, 5-Oct-2018.)

Hypotheses

Ref	Expression
ccatmulgnn0dir.a	|- A = ( ( 0..^ M ) X. { K } )
ccatmulgnn0dir.b	|- B = ( ( 0..^ N ) X. { K } )
ccatmulgnn0dir.c	|- C = ( ( 0..^ ( M + N ) ) X. { K } )
ccatmulgnn0dir.k	|- ( ph -> K e. S )
ccatmulgnn0dir.m	|- ( ph -> M e. NN0 )
ccatmulgnn0dir.n	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
ccatmulgnn0dir	|- ( ph -> ( A concat B ) = C )

Proof of Theorem ccatmulgnn0dir

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatmulgnn0dir.a	. . . . . . . . 9 |- A = ( ( 0..^ M ) X. { K } )
2	1	fveq2i 5860	. . . . . . . 8 |- ( # ` A ) = ( # ` ( ( 0..^ M ) X. { K } ) )
3		fzofi 12040	. . . . . . . . 9 |- ( 0..^ M ) e. Fin
4		snfi 7586	. . . . . . . . 9 |- { K } e. Fin
5		hashxp 12445	. . . . . . . . 9 |- ( ( ( 0..^ M ) e. Fin /\ { K } e. Fin ) -> ( # ` ( ( 0..^ M ) X. { K } ) ) = ( ( # ` ( 0..^ M ) ) x. ( # ` { K } ) ) )
6	3, 4, 5	mp2an 672	. . . . . . . 8 |- ( # ` ( ( 0..^ M ) X. { K } ) ) = ( ( # ` ( 0..^ M ) ) x. ( # ` { K } ) )
7	2, 6	eqtri 2489	. . . . . . 7 |- ( # ` A ) = ( ( # ` ( 0..^ M ) ) x. ( # ` { K } ) )
8		ccatmulgnn0dir.m	. . . . . . . . 9 |- ( ph -> M e. NN0 )
9		hashfzo0 12440	. . . . . . . . 9 |- ( M e. NN0 -> ( # ` ( 0..^ M ) ) = M )
10	8, 9	syl 16	. . . . . . . 8 |- ( ph -> ( # ` ( 0..^ M ) ) = M )
11		ccatmulgnn0dir.k	. . . . . . . . 9 |- ( ph -> K e. S )
12		hashsng 12393	. . . . . . . . 9 |- ( K e. S -> ( # ` { K } ) = 1 )
13	11, 12	syl 16	. . . . . . . 8 |- ( ph -> ( # ` { K } ) = 1 )
14	10, 13	oveq12d 6293	. . . . . . 7 |- ( ph -> ( ( # ` ( 0..^ M ) ) x. ( # ` { K } ) ) = ( M x. 1 ) )
15	7, 14	syl5eq 2513	. . . . . 6 |- ( ph -> ( # ` A ) = ( M x. 1 ) )
16	8	nn0cnd 10843	. . . . . . 7 |- ( ph -> M e. CC )
17	16	mulid1d 9602	. . . . . 6 |- ( ph -> ( M x. 1 ) = M )
18	15, 17	eqtrd 2501	. . . . 5 |- ( ph -> ( # ` A ) = M )
19		ccatmulgnn0dir.b	. . . . . . . . 9 |- B = ( ( 0..^ N ) X. { K } )
20	19	fveq2i 5860	. . . . . . . 8 |- ( # ` B ) = ( # ` ( ( 0..^ N ) X. { K } ) )
21		fzofi 12040	. . . . . . . . 9 |- ( 0..^ N ) e. Fin
22		hashxp 12445	. . . . . . . . 9 |- ( ( ( 0..^ N ) e. Fin /\ { K } e. Fin ) -> ( # ` ( ( 0..^ N ) X. { K } ) ) = ( ( # ` ( 0..^ N ) ) x. ( # ` { K } ) ) )
23	21, 4, 22	mp2an 672	. . . . . . . 8 |- ( # ` ( ( 0..^ N ) X. { K } ) ) = ( ( # ` ( 0..^ N ) ) x. ( # ` { K } ) )
24	20, 23	eqtri 2489	. . . . . . 7 |- ( # ` B ) = ( ( # ` ( 0..^ N ) ) x. ( # ` { K } ) )
25		ccatmulgnn0dir.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
26		hashfzo0 12440	. . . . . . . . 9 |- ( N e. NN0 -> ( # ` ( 0..^ N ) ) = N )
27	25, 26	syl 16	. . . . . . . 8 |- ( ph -> ( # ` ( 0..^ N ) ) = N )
28	27, 13	oveq12d 6293	. . . . . . 7 |- ( ph -> ( ( # ` ( 0..^ N ) ) x. ( # ` { K } ) ) = ( N x. 1 ) )
29	24, 28	syl5eq 2513	. . . . . 6 |- ( ph -> ( # ` B ) = ( N x. 1 ) )
30	25	nn0cnd 10843	. . . . . . 7 |- ( ph -> N e. CC )
31	30	mulid1d 9602	. . . . . 6 |- ( ph -> ( N x. 1 ) = N )
32	29, 31	eqtrd 2501	. . . . 5 |- ( ph -> ( # ` B ) = N )
33	18, 32	oveq12d 6293	. . . 4 |- ( ph -> ( ( # ` A ) + ( # ` B ) ) = ( M + N ) )
34	33	oveq2d 6291	. . 3 |- ( ph -> ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) = ( 0..^ ( M + N ) ) )
35		eqeq1 2464	. . . 4 |- ( ( A ` i ) = if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) -> ( ( A ` i ) = K <-> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) = K ) )
36		eqeq1 2464	. . . 4 |- ( ( B ` ( i - ( # ` A ) ) ) = if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) -> ( ( B ` ( i - ( # ` A ) ) ) = K <-> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) = K ) )
37		simpll 753	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ i e. ( 0..^ ( # ` A ) ) ) -> ph )
38		simpr 461	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ i e. ( 0..^ ( # ` A ) ) ) -> i e. ( 0..^ ( # ` A ) ) )
39	18	oveq2d 6291	. . . . . . 7 |- ( ph -> ( 0..^ ( # ` A ) ) = ( 0..^ M ) )
40	37, 39	syl 16	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ i e. ( 0..^ ( # ` A ) ) ) -> ( 0..^ ( # ` A ) ) = ( 0..^ M ) )
41	38, 40	eleqtrd 2550	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ i e. ( 0..^ ( # ` A ) ) ) -> i e. ( 0..^ M ) )
42		fconstg 5763	. . . . . . . 8 |- ( K e. S -> ( ( 0..^ M ) X. { K } ) : ( 0..^ M ) --> { K } )
43	11, 42	syl 16	. . . . . . 7 |- ( ph -> ( ( 0..^ M ) X. { K } ) : ( 0..^ M ) --> { K } )
44	1	a1i 11	. . . . . . . 8 |- ( ph -> A = ( ( 0..^ M ) X. { K } ) )
45	44	feq1d 5708	. . . . . . 7 |- ( ph -> ( A : ( 0..^ M ) --> { K } <-> ( ( 0..^ M ) X. { K } ) : ( 0..^ M ) --> { K } ) )
46	43, 45	mpbird 232	. . . . . 6 |- ( ph -> A : ( 0..^ M ) --> { K } )
47		fvconst 6070	. . . . . 6 |- ( ( A : ( 0..^ M ) --> { K } /\ i e. ( 0..^ M ) ) -> ( A ` i ) = K )
48	46, 47	sylan 471	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( A ` i ) = K )
49	37, 41, 48	syl2anc 661	. . . 4 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ i e. ( 0..^ ( # ` A ) ) ) -> ( A ` i ) = K )
50		simpll 753	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ph )
51		simplr 754	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) )
52		simpr 461	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> -. i e. ( 0..^ ( # ` A ) ) )
53	18, 8	eqeltrd 2548	. . . . . . . . 9 |- ( ph -> ( # ` A ) e. NN0 )
54	50, 53	syl 16	. . . . . . . 8 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( # ` A ) e. NN0 )
55	54	nn0zd 10953	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( # ` A ) e. ZZ )
56	32, 25	eqeltrd 2548	. . . . . . . . 9 |- ( ph -> ( # ` B ) e. NN0 )
57	50, 56	syl 16	. . . . . . . 8 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( # ` B ) e. NN0 )
58	57	nn0zd 10953	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( # ` B ) e. ZZ )
59		fzocatel 11837	. . . . . . 7 |- ( ( ( i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) /\ ( ( # ` A ) e. ZZ /\ ( # ` B ) e. ZZ ) ) -> ( i - ( # ` A ) ) e. ( 0..^ ( # ` B ) ) )
60	51, 52, 55, 58, 59	syl22anc 1224	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( i - ( # ` A ) ) e. ( 0..^ ( # ` B ) ) )
61	32	oveq2d 6291	. . . . . . 7 |- ( ph -> ( 0..^ ( # ` B ) ) = ( 0..^ N ) )
62	50, 61	syl 16	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( 0..^ ( # ` B ) ) = ( 0..^ N ) )
63	60, 62	eleqtrd 2550	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( i - ( # ` A ) ) e. ( 0..^ N ) )
64		fconstg 5763	. . . . . . . 8 |- ( K e. S -> ( ( 0..^ N ) X. { K } ) : ( 0..^ N ) --> { K } )
65	11, 64	syl 16	. . . . . . 7 |- ( ph -> ( ( 0..^ N ) X. { K } ) : ( 0..^ N ) --> { K } )
66	19	a1i 11	. . . . . . . 8 |- ( ph -> B = ( ( 0..^ N ) X. { K } ) )
67	66	feq1d 5708	. . . . . . 7 |- ( ph -> ( B : ( 0..^ N ) --> { K } <-> ( ( 0..^ N ) X. { K } ) : ( 0..^ N ) --> { K } ) )
68	65, 67	mpbird 232	. . . . . 6 |- ( ph -> B : ( 0..^ N ) --> { K } )
69		fvconst 6070	. . . . . 6 |- ( ( B : ( 0..^ N ) --> { K } /\ ( i - ( # ` A ) ) e. ( 0..^ N ) ) -> ( B ` ( i - ( # ` A ) ) ) = K )
70	68, 69	sylan 471	. . . . 5 |- ( ( ph /\ ( i - ( # ` A ) ) e. ( 0..^ N ) ) -> ( B ` ( i - ( # ` A ) ) ) = K )
71	50, 63, 70	syl2anc 661	. . . 4 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( B ` ( i - ( # ` A ) ) ) = K )
72	35, 36, 49, 71	ifbothda 3967	. . 3 |- ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) -> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) = K )
73	34, 72	mpteq12dva 4517	. 2 |- ( ph -> ( i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) ) = ( i e. ( 0..^ ( M + N ) ) |-> K ) )
74		ovex 6300	. . . . 5 |- ( 0..^ M ) e. _V
75		snex 4681	. . . . 5 |- { K } e. _V
76	74, 75	xpex 6704	. . . 4 |- ( ( 0..^ M ) X. { K } ) e. _V
77	1, 76	eqeltri 2544	. . 3 |- A e. _V
78		ovex 6300	. . . . 5 |- ( 0..^ N ) e. _V
79	78, 75	xpex 6704	. . . 4 |- ( ( 0..^ N ) X. { K } ) e. _V
80	19, 79	eqeltri 2544	. . 3 |- B e. _V
81		ccatfval 12544	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A concat B ) = ( i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) ) )
82	77, 80, 81	mp2an 672	. 2 |- ( A concat B ) = ( i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) )
83		ccatmulgnn0dir.c	. . 3 |- C = ( ( 0..^ ( M + N ) ) X. { K } )
84		fconstmpt 5035	. . 3 |- ( ( 0..^ ( M + N ) ) X. { K } ) = ( i e. ( 0..^ ( M + N ) ) |-> K )
85	83, 84	eqtri 2489	. 2 |- C = ( i e. ( 0..^ ( M + N ) ) |-> K )
86	73, 82, 85	3eqtr4g 2526	1 |- ( ph -> ( A concat B ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   _Vcvv 3106   ifcif 3932   {csn 4020   |-> cmpt 4498   X. cxp 4990   -->wf 5575   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481   1c1 9482   + caddc 9484   x. cmul 9486   - cmin 9794   NN0cn0 10784   ZZcz 10853  ..^cfzo 11781   #chash 12360   concat cconcat 12489

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-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-cda 8537  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-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-concat 12497

This theorem is referenced by:  ofcccat  28124