Theorem ccatw2s1p2 30441

Description: Extract the second of two single symbols concatenated with a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.)

Assertion

Ref	Expression
ccatw2s1p2	|- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W concat <" X "> ) concat <" Y "> ) ` ( N + 1 ) ) = Y )

Proof of Theorem ccatw2s1p2

Step	Hyp	Ref	Expression
1		simpl 457	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> W e. Word V )
2	1	anim1i 568	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( W e. Word V /\ ( X e. V /\ Y e. V ) ) )
3		3anass 969	. . . . 5 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) <-> ( W e. Word V /\ ( X e. V /\ Y e. V ) ) )
4	2, 3	sylibr 212	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( W e. Word V /\ X e. V /\ Y e. V ) )
5		ccatw2s1ass 12430	. . . 4 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W concat <" X "> ) concat <" Y "> ) = ( W concat ( <" X "> concat <" Y "> ) ) )
6	4, 5	syl 16	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W concat <" X "> ) concat <" Y "> ) = ( W concat ( <" X "> concat <" Y "> ) ) )
7	6	fveq1d 5804	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W concat <" X "> ) concat <" Y "> ) ` ( N + 1 ) ) = ( ( W concat ( <" X "> concat <" Y "> ) ) ` ( N + 1 ) ) )
8	1	adantr 465	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> W e. Word V )
9		ccat2s1cl 30434	. . . 4 |- ( ( X e. V /\ Y e. V ) -> ( <" X "> concat <" Y "> ) e. Word V )
10	9	adantl 466	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( <" X "> concat <" Y "> ) e. Word V )
11		oveq1 6210	. . . . . . . 8 |- ( N = ( # ` W ) -> ( N + 1 ) = ( ( # ` W ) + 1 ) )
12	11	eqcoms 2466	. . . . . . 7 |- ( ( # ` W ) = N -> ( N + 1 ) = ( ( # ` W ) + 1 ) )
13	12	adantl 466	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( N + 1 ) = ( ( # ` W ) + 1 ) )
14		lencl 12371	. . . . . . . 8 |- ( W e. Word V -> ( # ` W ) e. NN0 )
15		nn0z 10784	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
16		1nn0 10710	. . . . . . . . . 10 |- 1 e. NN0
17		zpnn0elfzo1 30393	. . . . . . . . . 10 |- ( ( ( # ` W ) e. ZZ /\ 1 e. NN0 ) -> ( ( # ` W ) + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + ( 1 + 1 ) ) ) )
18	15, 16, 17	sylancl 662	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + ( 1 + 1 ) ) ) )
19		1p1e2 10550	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
20	19	oveq2i 6214	. . . . . . . . . . 11 |- ( ( # ` W ) + ( 1 + 1 ) ) = ( ( # ` W ) + 2 )
21	20	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) + ( 1 + 1 ) ) = ( ( # ` W ) + 2 ) )
22	21	oveq2d 6219	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W )..^ ( ( # ` W ) + ( 1 + 1 ) ) ) = ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
23	18, 22	eleqtrd 2544	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
24	14, 23	syl 16	. . . . . . 7 |- ( W e. Word V -> ( ( # ` W ) + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
25	24	adantr 465	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( ( # ` W ) + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
26	13, 25	eqeltrd 2542	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( N + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
27	26	adantr 465	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( N + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
28		ccat2s1len 30435	. . . . . . 7 |- ( ( X e. V /\ Y e. V ) -> ( # ` ( <" X "> concat <" Y "> ) ) = 2 )
29	28	adantl 466	. . . . . 6 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( # ` ( <" X "> concat <" Y "> ) ) = 2 )
30	29	oveq2d 6219	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( # ` W ) + ( # ` ( <" X "> concat <" Y "> ) ) ) = ( ( # ` W ) + 2 ) )
31	30	oveq2d 6219	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( # ` W )..^ ( ( # ` W ) + ( # ` ( <" X "> concat <" Y "> ) ) ) ) = ( ( # ` W )..^ ( ( # ` W ) + 2 ) ) )
32	27, 31	eleqtrrd 2545	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( N + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + ( # ` ( <" X "> concat <" Y "> ) ) ) ) )
33		ccatval2 12399	. . 3 |- ( ( W e. Word V /\ ( <" X "> concat <" Y "> ) e. Word V /\ ( N + 1 ) e. ( ( # ` W )..^ ( ( # ` W ) + ( # ` ( <" X "> concat <" Y "> ) ) ) ) ) -> ( ( W concat ( <" X "> concat <" Y "> ) ) ` ( N + 1 ) ) = ( ( <" X "> concat <" Y "> ) ` ( ( N + 1 ) - ( # ` W ) ) ) )
34	8, 10, 32, 33	syl3anc 1219	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W concat ( <" X "> concat <" Y "> ) ) ` ( N + 1 ) ) = ( ( <" X "> concat <" Y "> ) ` ( ( N + 1 ) - ( # ` W ) ) ) )
35	12	oveq1d 6218	. . . . . 6 |- ( ( # ` W ) = N -> ( ( N + 1 ) - ( # ` W ) ) = ( ( ( # ` W ) + 1 ) - ( # ` W ) ) )
36		nn0cn 10704	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
37		ax-1cn 9455	. . . . . . . 8 |- 1 e. CC
38	37	a1i 11	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> 1 e. CC )
39	36, 38	pncan2d 9836	. . . . . 6 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) + 1 ) - ( # ` W ) ) = 1 )
40	35, 39	sylan9eqr 2517	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ ( # ` W ) = N ) -> ( ( N + 1 ) - ( # ` W ) ) = 1 )
41	14, 40	sylan 471	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( ( N + 1 ) - ( # ` W ) ) = 1 )
42	41	fveq2d 5806	. . 3 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( ( <" X "> concat <" Y "> ) ` ( ( N + 1 ) - ( # ` W ) ) ) = ( ( <" X "> concat <" Y "> ) ` 1 ) )
43		ccat2s1p2 30437	. . 3 |- ( ( X e. V /\ Y e. V ) -> ( ( <" X "> concat <" Y "> ) ` 1 ) = Y )
44	42, 43	sylan9eq 2515	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( <" X "> concat <" Y "> ) ` ( ( N + 1 ) - ( # ` W ) ) ) = Y )
45	7, 34, 44	3eqtrd 2499	1 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W concat <" X "> ) concat <" Y "> ) ` ( N + 1 ) ) = Y )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9395   1c1 9398   + caddc 9400   - cmin 9710   2c2 10486   NN0cn0 10694   ZZcz 10761  ..^cfzo 11669   #chash 12224  Word cword 12343   concat cconcat 12345   <"cs1 12346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-concat 12353  df-s1 12354

This theorem is referenced by:  numclwwlkovf2ex  30850  numclwlk1lem2foa  30855