Theorem repswccat 12769

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 12760	. . . . . 6 |- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
2	1	3adant3 1016	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
3		repswlen 12760	. . . . . 6 |- ( ( S e. V /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
4	3	3adant2 1015	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
5	2, 4	oveq12d 6314	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
6	5	oveq2d 6312	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
7		simp1 996	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> S e. V )
8	7	adantr 465	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> S e. V )
9		simpl2 1000	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> N e. NN0 )
10	2	oveq2d 6312	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
11	10	eleq2d 2527	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
12	11	biimpa 484	. . . . . . 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 1176	. . . . . 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 714	. . . . 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 12758	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ x e. ( 0..^ N ) ) -> ( ( S repeatS N ) ` x ) = S )
16	14, 15	syl 16	. . . 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 725	. . . . 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 1037	. . . . 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 532	. . . . . . 7 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) )
20		simpr 461	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) -> x e. ( 0..^ ( N + M ) ) )
21	20	anim1i 568	. . . . . . . . . . 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 10908	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N e. ZZ )
23		nn0z 10908	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
24	22, 23	anim12i 566	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N e. ZZ /\ M e. ZZ ) )
25	24	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( N e. ZZ /\ M e. ZZ ) )
26		fzocatel 11883	. . . . . . . . . . 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 661	. . . . . . . . . 10 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( x - N ) e. ( 0..^ M ) )
28	27	exp31 604	. . . . . . . . 9 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
29	28	3adant1 1014	. . . . . . . 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 6305	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
31	30	oveq2d 6312	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
32	31	eleq2d 2527	. . . . . . . . 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 6304	. . . . . . . . . . . . 13 |- ( ( # ` ( S repeatS N ) ) = N -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
34	33	eleq2d 2527	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
35	34	notbid 294	. . . . . . . . . . 11 |- ( ( # ` ( S repeatS N ) ) = N -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
36	35	adantr 465	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
37		oveq2 6304	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( x - ( # ` ( S repeatS N ) ) ) = ( x - N ) )
38	37	eleq1d 2526	. . . . . . . . . . 11 |- ( ( # ` ( S repeatS N ) ) = N -> ( ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) <-> ( x - N ) e. ( 0..^ M ) ) )
39	38	adantr 465	. . . . . . . . . 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 320	. . . . . . . . 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 320	. . . . . . . 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 221	. . . . . . 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 36	. . . . . 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 432	. . . . 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 12758	. . . . 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 1228	. . . 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 3977	. . 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 4534	. 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 6324	. . . 4 |- ( S repeatS N ) e. _V
50		ovex 6324	. . . 4 |- ( S repeatS M ) e. _V
51	49, 50	pm3.2i 455	. . 3 |- ( ( S repeatS N ) e. _V /\ ( S repeatS M ) e. _V )
52		ccatfval 12601	. . 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 12	. 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 10852	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
55	54	3adant1 1014	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
56		reps 12754	. . 3 |- ( ( S e. V /\ ( N + M ) e. NN0 ) -> ( S repeatS ( N + M ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
57	7, 55, 56	syl2anc 661	. 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 2508	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 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   _Vcvv 3109   ifcif 3944   |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   0cc0 9509   + caddc 9512   - cmin 9824   NN0cn0 10816   ZZcz 10885  ..^cfzo 11821   #chash 12408   ++ cconcat 12540   repeatS creps 12545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-concat 12548  df-reps 12553

This theorem is referenced by:  repswcshw  12792  repsw2  12900  repsw3  12901