Theorem ccatmulgnn0dir 26959

Description: Concatenation of words follow the rule mulgnn0dir 15669 (although applying mulgnn0dir 15669 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 5713	. . . . . . . 8 |- ( # ` A ) = ( # ` ( ( 0..^ M ) X. { K } ) )
3		fzofi 11815	. . . . . . . . 9 |- ( 0..^ M ) e. Fin
4		snfi 7409	. . . . . . . . 9 |- { K } e. Fin
5		hashxp 12215	. . . . . . . . 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 2463	. . . . . . 7 |- ( # ` A ) = ( ( # ` ( 0..^ M ) ) x. ( # ` { K } ) )
8		ccatmulgnn0dir.m	. . . . . . . . 9 |- ( ph -> M e. NN0 )
9		hashfzo0 12210	. . . . . . . . 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 12155	. . . . . . . . 9 |- ( K e. S -> ( # ` { K } ) = 1 )
13	11, 12	syl 16	. . . . . . . 8 |- ( ph -> ( # ` { K } ) = 1 )
14	10, 13	oveq12d 6128	. . . . . . 7 |- ( ph -> ( ( # ` ( 0..^ M ) ) x. ( # ` { K } ) ) = ( M x. 1 ) )
15	7, 14	syl5eq 2487	. . . . . 6 |- ( ph -> ( # ` A ) = ( M x. 1 ) )
16	8	nn0cnd 10657	. . . . . . 7 |- ( ph -> M e. CC )
17	16	mulid1d 9422	. . . . . 6 |- ( ph -> ( M x. 1 ) = M )
18	15, 17	eqtrd 2475	. . . . 5 |- ( ph -> ( # ` A ) = M )
19		ccatmulgnn0dir.b	. . . . . . . . 9 |- B = ( ( 0..^ N ) X. { K } )
20	19	fveq2i 5713	. . . . . . . 8 |- ( # ` B ) = ( # ` ( ( 0..^ N ) X. { K } ) )
21		fzofi 11815	. . . . . . . . 9 |- ( 0..^ N ) e. Fin
22		hashxp 12215	. . . . . . . . 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 2463	. . . . . . 7 |- ( # ` B ) = ( ( # ` ( 0..^ N ) ) x. ( # ` { K } ) )
25		ccatmulgnn0dir.n	. . . . . . . . 9 |- ( ph -> N e. NN0 )
26		hashfzo0 12210	. . . . . . . . 9 |- ( N e. NN0 -> ( # ` ( 0..^ N ) ) = N )
27	25, 26	syl 16	. . . . . . . 8 |- ( ph -> ( # ` ( 0..^ N ) ) = N )
28	27, 13	oveq12d 6128	. . . . . . 7 |- ( ph -> ( ( # ` ( 0..^ N ) ) x. ( # ` { K } ) ) = ( N x. 1 ) )
29	24, 28	syl5eq 2487	. . . . . 6 |- ( ph -> ( # ` B ) = ( N x. 1 ) )
30	25	nn0cnd 10657	. . . . . . 7 |- ( ph -> N e. CC )
31	30	mulid1d 9422	. . . . . 6 |- ( ph -> ( N x. 1 ) = N )
32	29, 31	eqtrd 2475	. . . . 5 |- ( ph -> ( # ` B ) = N )
33	18, 32	oveq12d 6128	. . . 4 |- ( ph -> ( ( # ` A ) + ( # ` B ) ) = ( M + N ) )
34	33	oveq2d 6126	. . 3 |- ( ph -> ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) = ( 0..^ ( M + N ) ) )
35		eqeq1 2449	. . . 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 2449	. . . 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 6126	. . . . . . 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 2519	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ i e. ( 0..^ ( # ` A ) ) ) -> i e. ( 0..^ M ) )
42		fconstg 5616	. . . . . . . 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 5565	. . . . . . 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 5916	. . . . . 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 2517	. . . . . . . . 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 10764	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( # ` A ) e. ZZ )
56	32, 25	eqeltrd 2517	. . . . . . . . 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 10764	. . . . . . 7 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( # ` B ) e. ZZ )
59		fzocatel 11621	. . . . . . 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 1219	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( i - ( # ` A ) ) e. ( 0..^ ( # ` B ) ) )
61	32	oveq2d 6126	. . . . . . 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 2519	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) /\ -. i e. ( 0..^ ( # ` A ) ) ) -> ( i - ( # ` A ) ) e. ( 0..^ N ) )
64		fconstg 5616	. . . . . . . 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 5565	. . . . . . 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 5916	. . . . . 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 3843	. . 3 |- ( ( ph /\ i e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) -> if ( i e. ( 0..^ ( # ` A ) ) , ( A ` i ) , ( B ` ( i - ( # ` A ) ) ) ) = K )
73	34, 72	mpteq12dva 4388	. 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 6135	. . . . 5 |- ( 0..^ M ) e. _V
75		snex 4552	. . . . 5 |- { K } e. _V
76	74, 75	xpex 6527	. . . 4 |- ( ( 0..^ M ) X. { K } ) e. _V
77	1, 76	eqeltri 2513	. . 3 |- A e. _V
78		ovex 6135	. . . . 5 |- ( 0..^ N ) e. _V
79	78, 75	xpex 6527	. . . 4 |- ( ( 0..^ N ) X. { K } ) e. _V
80	19, 79	eqeltri 2513	. . 3 |- B e. _V
81		ccatfval 12292	. . 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 4901	. . 3 |- ( ( 0..^ ( M + N ) ) X. { K } ) = ( i e. ( 0..^ ( M + N ) ) |-> K )
85	83, 84	eqtri 2463	. 2 |- C = ( i e. ( 0..^ ( M + N ) ) |-> K )
86	73, 82, 85	3eqtr4g 2500	1 |- ( ph -> ( A concat B ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2991   ifcif 3810   {csn 3896   e. cmpt 4369   X. cxp 4857   -->wf 5433   ` cfv 5437  (class class class)co 6110   Fincfn 7329   0cc0 9301   1c1 9302   + caddc 9304   x. cmul 9306   - cmin 9614   NN0cn0 10598   ZZcz 10665  ..^cfzo 11567   #chash 12122   concat cconcat 12242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-cda 8356  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-n0 10599  df-z 10666  df-uz 10881  df-fz 11457  df-fzo 11568  df-hash 12123  df-concat 12250

This theorem is referenced by:  ofcccat  26961