Theorem repswccat 12883

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 12874	. . . . . 6 |- ( ( S e. V /\ N e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
2	1	3adant3 1027	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS N ) ) = N )
3		repswlen 12874	. . . . . 6 |- ( ( S e. V /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
4	3	3adant2 1026	. . . . 5 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( # ` ( S repeatS M ) ) = M )
5	2, 4	oveq12d 6306	. . . 4 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
6	5	oveq2d 6304	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
7		simp1 1007	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> S e. V )
8	7	adantr 467	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> S e. V )
9		simpl2 1011	. . . . . . 7 |- ( ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( # ` ( S repeatS N ) ) ) ) -> N e. NN0 )
10	2	oveq2d 6304	. . . . . . . . 9 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
11	10	eleq2d 2513	. . . . . . . 8 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
12	11	biimpa 487	. . . . . . 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 1187	. . . . . 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 720	. . . . 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 12872	. . . . 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 731	. . . . 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 1048	. . . . 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 535	. . . . . . 7 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) )
20		simpr 463	. . . . . . . . . . . 12 |- ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) -> x e. ( 0..^ ( N + M ) ) )
21	20	anim1i 571	. . . . . . . . . . 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 10957	. . . . . . . . . . . . 13 |- ( N e. NN0 -> N e. ZZ )
23		nn0z 10957	. . . . . . . . . . . . 13 |- ( M e. NN0 -> M e. ZZ )
24	22, 23	anim12i 569	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N e. ZZ /\ M e. ZZ ) )
25	24	ad2antrr 731	. . . . . . . . . . 11 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( N e. ZZ /\ M e. ZZ ) )
26		fzocatel 11975	. . . . . . . . . . 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 666	. . . . . . . . . 10 |- ( ( ( ( N e. NN0 /\ M e. NN0 ) /\ x e. ( 0..^ ( N + M ) ) ) /\ -. x e. ( 0..^ N ) ) -> ( x - N ) e. ( 0..^ M ) )
28	27	exp31 608	. . . . . . . . 9 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( x e. ( 0..^ ( N + M ) ) -> ( -. x e. ( 0..^ N ) -> ( x - N ) e. ( 0..^ M ) ) ) )
29	28	3adant1 1025	. . . . . . . 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 6297	. . . . . . . . . . 11 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) = ( N + M ) )
31	30	oveq2d 6304	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( 0..^ ( ( # ` ( S repeatS N ) ) + ( # ` ( S repeatS M ) ) ) ) = ( 0..^ ( N + M ) ) )
32	31	eleq2d 2513	. . . . . . . . 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 6296	. . . . . . . . . . . . 13 |- ( ( # ` ( S repeatS N ) ) = N -> ( 0..^ ( # ` ( S repeatS N ) ) ) = ( 0..^ N ) )
34	33	eleq2d 2513	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> x e. ( 0..^ N ) ) )
35	34	notbid 296	. . . . . . . . . . 11 |- ( ( # ` ( S repeatS N ) ) = N -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
36	35	adantr 467	. . . . . . . . . 10 |- ( ( ( # ` ( S repeatS N ) ) = N /\ ( # ` ( S repeatS M ) ) = M ) -> ( -. x e. ( 0..^ ( # ` ( S repeatS N ) ) ) <-> -. x e. ( 0..^ N ) ) )
37		oveq2 6296	. . . . . . . . . . . 12 |- ( ( # ` ( S repeatS N ) ) = N -> ( x - ( # ` ( S repeatS N ) ) ) = ( x - N ) )
38	37	eleq1d 2512	. . . . . . . . . . 11 |- ( ( # ` ( S repeatS N ) ) = N -> ( ( x - ( # ` ( S repeatS N ) ) ) e. ( 0..^ M ) <-> ( x - N ) e. ( 0..^ M ) ) )
39	38	adantr 467	. . . . . . . . . 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 322	. . . . . . . . 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 322	. . . . . . . 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 225	. . . . . . 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 37	. . . . . 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 434	. . . . 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 12872	. . . . 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 1267	. . . 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 3913	. . 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 4479	. 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 6316	. . . 4 |- ( S repeatS N ) e. _V
50		ovex 6316	. . . 4 |- ( S repeatS M ) e. _V
51	49, 50	pm3.2i 457	. . 3 |- ( ( S repeatS N ) e. _V /\ ( S repeatS M ) e. _V )
52		ccatfval 12716	. . 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 10902	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
55	54	3adant1 1025	. . 3 |- ( ( S e. V /\ N e. NN0 /\ M e. NN0 ) -> ( N + M ) e. NN0 )
56		reps 12868	. . 3 |- ( ( S e. V /\ ( N + M ) e. NN0 ) -> ( S repeatS ( N + M ) ) = ( x e. ( 0..^ ( N + M ) ) |-> S ) )
57	7, 55, 56	syl2anc 666	. 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 2494	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 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   _Vcvv 3044   ifcif 3880   |-> cmpt 4460   ` cfv 5581  (class class class)co 6288   0cc0 9536   + caddc 9539   - cmin 9857   NN0cn0 10866   ZZcz 10934  ..^cfzo 11912   #chash 12512   ++ cconcat 12655   repeatS creps 12660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-hash 12513  df-concat 12663  df-reps 12668

This theorem is referenced by:  repswcshw  12906  repsw2  13018  repsw3  13019