Theorem repswccat 12423

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 12414	. . . . . 6 |- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
2	1	3adant3 1008	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
3		repswlen 12414	. . . . . 6 |- ( ( S e. V /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
4	3	3adant2 1007	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
5	2, 4	oveq12d 6109	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
6	5	oveq2d 6107	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
7		simp1 988	. . . . . . . 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 992	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> N e. NN0 )
10	2	oveq2d 6107	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
11	10	eleq2d 2510	. . . . . . . 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 1168	. . . . . 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 12412	. . . . 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 1029	. . . . 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 10669	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> N e. ZZ )
23		nn0z 10669	. . . . . . . . . . . . . . 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 11602	. . . . . . . . . . . . 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 1006	. . . . . . . . 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 6100	. . . . . . . . . . . 12 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
32	31	oveq2d 6107	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
33	32	eleq2d 2510	. . . . . . . . . 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 6099	. . . . . . . . . . . . . 14 |- ( ( # ` ( S repeatS N ) ) = N -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
35	34	eleq2d 2510	. . . . . . . . . . . . 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 6099	. . . . . . . . . . . . 13 |- ( ( # ` ( S repeatS N ) ) = N -> ( x - ( # ` ( S repeatS N ) ) ) = ( x - N ) )
39	38	eleq1d 2509	. . . . . . . . . . . 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 12412	. . . . 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 1218	. . . 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 3822	. . 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 4369	. 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 6116	. . . 4 |- ( S repeatS N ) e. _V
52		ovex 6116	. . . 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 12273	. . 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 10615	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
57	56	3adant1 1006	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
58		reps 12408	. . 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 2485	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 965   = wceq 1369   e. wcel 1756   _Vcvv 2972   ifcif 3791   e. cmpt 4350   ` cfv 5418  (class class class)co 6091   0cc0 9282   + caddc 9285   - cmin 9595   NN0cn0 10579   ZZcz 10646  ..^cfzo 11548   #chash 12103   concat cconcat 12223   repeatS creps 12228

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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-concat 12231  df-reps 12236

This theorem is referenced by:  repswcshw  12446  repsw2  12550  repsw3  12551