Theorem repswccat 13532

Description: The concatenation of two "repeated symbol words" with the same symbol is again a "repeated symbol word". (Contributed by AV, 4-Nov-2018.)

Assertion

Ref	Expression
repswccat	|- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( S repeatS N ) ++ ( S repeatS M ) ) = ( S repeatS ( N + M ) ) )

Proof of Theorem repswccat

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		repswlen 13523	. . . . . 6 |- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
2	1	3adant3 1081	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
3		repswlen 13523	. . . . . 6 |- ( ( S e. V /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
4	3	3adant2 1080	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
5	2, 4	oveq12d 6668	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
6	5	oveq2d 6666	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
7		simp1 1061	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> S e. V )
8	7	adantr 481	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> S e. V )
9		simpl2 1065	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> N e. NN0 )
10	2	oveq2d 6666	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
11	10	eleq2d 2687	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
12	11	biimpa 501	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> x e. ( 0..^ N ) )
13	8, 9, 12	3jca 1242	. . . . . 6 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> ( S e. V /\ N e. NN0 /\ x e. ( 0..^ N ) ) )
14	13	adantlr 751	. . . . 5 |- ( ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> ( S e. V /\ N e. NN0 /\ x e. ( 0..^ N ) ) )
15		repswsymb 13521	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ x e. ( 0..^ N ) ) -> ( ( S repeatS N ) ` x ) = S )
16	14, 15	syl 17	. . . 4 |- ( ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> ( ( S repeatS N ) ` x ) = S )
17	7	ad2antrr 762	. . . . 5 |- ( ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) /\ -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> S e. V )
18		simpll3 1102	. . . . 5 |- ( ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) /\ -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> M e. NN0 )
19	2, 4	jca 554	. . . . . . 7 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) )
20		simpr 477	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) -> x e. ( 0..^ ( N + M ) ) )
21	20	anim1i 592	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( x e. ( 0..^ ( N + M ) ) /\ -. x e. ( 0..^ N ) ) )
22		nn0z 11400	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N e. ZZ )
23		nn0z 11400	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
24	22, 23	anim12i 590	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N e. ZZ /\ M e. ZZ ) )
25	24	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( N e. ZZ /\ M e. ZZ ) )
26		fzocatel 12531	. . . . . . . . . . 11 |- ( ( ( x e. ( 0..^ ( N + M ) ) /\ -. x e. ( 0..^ N ) ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( x - N ) e. ( 0..^ M ) )
27	21, 25, 26	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( x - N ) e. ( 0..^ M ) )
28	27	exp31 630	. . . . . . . . 9 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
29	28	3adant1 1079	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
30		oveq12 6659	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
31	30	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
32	31	eleq2d 2687	. . . . . . . . 9 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) <-> x e. ( 0..^ ( N + M ) ) ) )
33		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` ( S repeatS N ) ) = N -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
34	33	eleq2d 2687	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
35	34	notbid 308	. . . . . . . . . . 11 |- ( ( # ` ( S repeatS N ) ) = N -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
36	35	adantr 481	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
37		oveq2 6658	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( x - ( # ` ( S repeatS N ) ) ) = ( x - N ) )
38	37	eleq1d 2686	. . . . . . . . . . 11 |- ( ( # ` ( S repeatS N ) ) = N -> ( ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) <-> ( x - N ) e. ( 0..^ M ) ) )
39	38	adantr 481	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) <-> ( x - N ) e. ( 0..^ M ) ) )
40	36, 39	imbi12d 334	. . . . . . . . 9 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) -> ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) ) <-> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
41	32, 40	imbi12d 334	. . . . . . . 8 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) -> ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) ) ) <-> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) ) )
42	29, 41	syl5ibr 236	. . . . . . 7 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) -> ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) ) ) ) )
43	19, 42	mpcom 38	. . . . . 6 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) -> ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) ) ) )
44	43	imp31 448	. . . . 5 |- ( ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) /\ -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) )
45		repswsymb 13521	. . . . 5 |- ( ( S e. V /\ M e. NN0 /\ ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) ) -> ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) = S )
46	17, 18, 44, 45	syl3anc 1326	. . . 4 |- ( ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) /\ -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) = S )
47	16, 46	ifeqda 4121	. . 3 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) ) -> if ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) , ( ( S repeatS N ) ` x ) , ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) ) = S )
48	6, 47	mpteq12dva 4732	. 2 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) |-> if ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) , ( ( S repeatS N ) ` x ) , ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
49		ovex 6678	. . . 4 |- ( S repeatS N ) e. _V
50		ovex 6678	. . . 4 |- ( S repeatS M ) e. _V
51	49, 50	pm3.2i 471	. . 3 |- ( ( S repeatS N ) e. _V /\ ( S repeatS M ) e. _V )
52		ccatfval 13358	. . 3 |- ( ( ( S repeatS N ) e. _V /\ ( S repeatS M ) e. _V ) -> ( ( S repeatS N ) ++ ( S repeatS M ) ) = ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) |-> if ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) , ( ( S repeatS N ) ` x ) , ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) ) ) )
53	51, 52	mp1i 13	. 2 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( S repeatS N ) ++ ( S repeatS M ) ) = ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) |-> if ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) , ( ( S repeatS N ) ` x ) , ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) ) ) )
54		nn0addcl 11328	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
55	54	3adant1 1079	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
56		reps 13517	. . 3 |- ( ( S e. V /\ ( N + M ) e. NN0 ) -> ( S repeatS ( N + M ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
57	7, 55, 56	syl2anc 693	. 2 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( S repeatS ( N + M ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
58	48, 53, 57	3eqtr4d 2666	1 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( S repeatS N ) ++ ( S repeatS M ) ) = ( S repeatS ( N + M ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   #chash 13117   ++ cconcat 13293   repeatS creps 13298

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-concat 13301  df-reps 13306

This theorem is referenced by:  repswcshw  13558  repsw2  13693  repsw3  13694