Theorem repswccat 12732

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 ) concat ( 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 12723	. . . . . 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 12723	. . . . . 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 6312	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
6	5	oveq2d 6310	. . 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 6310	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
11	10	eleq2d 2537	. . . . . . . 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 12721	. . . . 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	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) )
20		simpr 461	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) -> x e. ( 0..^ ( N + M ) ) )
21	20	anim1i 568	. . . . . . . . . . . . 13 |- ( ( ( ( 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 10897	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
23		nn0z 10897	. . . . . . . . . . . . . . 15 |- ( M e. NN0 -> M e. ZZ )
24	22, 23	anim12i 566	. . . . . . . . . . . . . 14 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N e. ZZ /\ M e. ZZ ) )
25	24	ad2antrr 725	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( N e. ZZ /\ M e. ZZ ) )
26		fzocatel 11858	. . . . . . . . . . . . 13 |- ( ( ( 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	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( x - N ) e. ( 0..^ M ) )
28	27	ex 434	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) )
29	28	ex 434	. . . . . . . . . 10 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
30	29	3adant1 1014	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
31		oveq12 6303	. . . . . . . . . . . 12 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
32	31	oveq2d 6310	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
33	32	eleq2d 2537	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( x e. ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) <-> x e. ( 0..^ ( N + M ) ) ) )
34		oveq2 6302	. . . . . . . . . . . . . 14 |- ( ( # ` ( S repeatS N ) ) = N -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
35	34	eleq2d 2537	. . . . . . . . . . . . 13 |- ( ( # ` ( S repeatS N ) ) = N -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
36	35	notbid 294	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
37	36	adantr 465	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
38		oveq2 6302	. . . . . . . . . . . . 13 |- ( ( # ` ( S repeatS N ) ) = N -> ( x - ( # ` ( S repeatS N ) ) ) = ( x - N ) )
39	38	eleq1d 2536	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) <-> ( x - N ) e. ( 0..^ M ) ) )
40	39	adantr 465	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) <-> ( x - N ) e. ( 0..^ M ) ) )
41	37, 40	imbi12d 320	. . . . . . . . . 10 |- ( ( ( # ` ( 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 ) ) ) )
42	33, 41	imbi12d 320	. . . . . . . . 9 |- ( ( ( # ` ( 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 ) ) ) ) )
43	30, 42	syl5ibr 221	. . . . . . . 8 |- ( ( ( # ` ( 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 ) ) ) ) )
44	19, 43	mpcom 36	. . . . . . 7 |- ( ( 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	44	imp 429	. . . . . 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 ) ) )
46	45	imp 429	. . . . 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 ) )
47		repswsymb 12721	. . . . 5 |- ( ( S e. V /\ M e. NN0 /\ ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) ) -> ( ( S repeatS M ) ` ( x - ( # ` ( S repeatS N ) ) ) ) = S )
48	17, 18, 46, 47	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 )
49	16, 48	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 )
50	6, 49	mpteq12dva 4529	. 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 ) )
51		ovex 6319	. . . 4 |- ( S repeatS N ) e. _V
52		ovex 6319	. . . 4 |- ( S repeatS M ) e. _V
53	51, 52	pm3.2i 455	. . 3 |- ( ( S repeatS N ) e. _V /\ ( S repeatS M ) e. _V )
54		ccatfval 12567	. . 3 |- ( ( ( S repeatS N ) e. _V /\ ( S repeatS M ) e. _V ) -> ( ( S repeatS N ) concat ( 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 ) ) ) ) ) ) )
55	53, 54	mp1i 12	. 2 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( S repeatS N ) concat ( 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 ) ) ) ) ) ) )
56		nn0addcl 10841	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
57	56	3adant1 1014	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
58		reps 12717	. . 3 |- ( ( S e. V /\ ( N + M ) e. NN0 ) -> ( S repeatS ( N + M ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
59	7, 57, 58	syl2anc 661	. 2 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( S repeatS ( N + M ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
60	50, 55, 59	3eqtr4d 2518	1 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( S repeatS N ) concat ( S repeatS M ) ) = ( S repeatS ( N + M ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   _Vcvv 3118   ifcif 3944   |-> cmpt 4510   ` cfv 5593  (class class class)co 6294   0cc0 9502   + caddc 9505   - cmin 9815   NN0cn0 10805   ZZcz 10874  ..^cfzo 11802   #chash 12383   concat cconcat 12512   repeatS creps 12517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-hash 12384  df-concat 12520  df-reps 12525

This theorem is referenced by:  repswcshw  12755  repsw2  12863  repsw3  12864