Theorem swrd2lsw 12865

Description: Extract the last two single symbol from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)

Assertion

Ref	Expression
swrd2lsw	|- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )

Proof of Theorem swrd2lsw

Step	Hyp	Ref	Expression
1		simpl 457	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> W e. Word V )
2		lencl 12538	. . . . 5 |- ( W e. Word V -> ( # ` W ) e. NN0 )
3		1z 10904	. . . . . . . . 9 |- 1 e. ZZ
4		nn0z 10897	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
5		zltp1le 10922	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ ( # ` W ) e. ZZ ) -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
6	3, 4, 5	sylancr 663	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
7		1p1e2 10659	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
8	7	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 1 + 1 ) = 2 )
9	8	breq1d 4462	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) <-> 2 <_ ( # ` W ) ) )
10	9	biimpd 207	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) -> 2 <_ ( # ` W ) ) )
11	6, 10	sylbid 215	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> 2 <_ ( # ` W ) ) )
12	11	imp 429	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> 2 <_ ( # ` W ) )
13		2nn0 10822	. . . . . . . . 9 |- 2 e. NN0
14	13	jctl 541	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) )
15	14	adantr 465	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) )
16		nn0sub 10856	. . . . . . 7 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) -> ( 2 <_ ( # ` W ) <-> ( ( # ` W ) - 2 ) e. NN0 ) )
17	15, 16	syl 16	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( 2 <_ ( # ` W ) <-> ( ( # ` W ) - 2 ) e. NN0 ) )
18	12, 17	mpbid 210	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
19	2, 18	sylan 471	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
20		0red 9607	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 0 e. RR )
21		1red 9621	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 1 e. RR )
22		zre 10878	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> ( # ` W ) e. RR )
23	20, 21, 22	3jca 1176	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) )
24		0lt1 10085	. . . . . . . . . . 11 |- 0 < 1
25		lttr 9671	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( # ` W ) ) -> 0 < ( # ` W ) ) )
26	25	expd 436	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( 0 < 1 -> ( 1 < ( # ` W ) -> 0 < ( # ` W ) ) ) )
27	23, 24, 26	mpisyl 18	. . . . . . . . . 10 |- ( ( # ` W ) e. ZZ -> ( 1 < ( # ` W ) -> 0 < ( # ` W ) ) )
28		elnnz 10884	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. ZZ /\ 0 < ( # ` W ) ) )
29	28	simplbi2 625	. . . . . . . . . 10 |- ( ( # ` W ) e. ZZ -> ( 0 < ( # ` W ) -> ( # ` W ) e. NN ) )
30	27, 29	syld 44	. . . . . . . . 9 |- ( ( # ` W ) e. ZZ -> ( 1 < ( # ` W ) -> ( # ` W ) e. NN ) )
31	4, 30	syl 16	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> ( # ` W ) e. NN ) )
32	31	imp 429	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( # ` W ) e. NN )
33		fzo0end 11882	. . . . . . 7 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
34	32, 33	syl 16	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
35		nn0cn 10815	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
36		2cn 10616	. . . . . . . . . . . 12 |- 2 e. CC
37	36	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 2 e. CC )
38		ax-1cn 9560	. . . . . . . . . . . 12 |- 1 e. CC
39	38	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 1 e. CC )
40	35, 37, 39	3jca 1176	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) )
41		1e2m1 10661	. . . . . . . . . . . . 13 |- 1 = ( 2 - 1 )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> 1 = ( 2 - 1 ) )
43	42	oveq2d 6310	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
44		subsub 9859	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
45	43, 44	eqtrd 2508	. . . . . . . . . 10 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
46	40, 45	syl 16	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
47	46	eqcomd 2475	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
48	47	eleq1d 2536	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
49	48	adantr 465	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
50	34, 49	mpbird 232	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
51	2, 50	sylan 471	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
52	1, 19, 51	3jca 1176	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W e. Word V /\ ( ( # ` W ) - 2 ) e. NN0 /\ ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) ) )
53		swrds2 12858	. . 3 |- ( ( W e. Word V /\ ( ( # ` W ) - 2 ) e. NN0 /\ ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
54	52, 53	syl 16	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
55	35, 36	jctir 538	. . . . . 6 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC ) )
56		npcan 9839	. . . . . . 7 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( ( ( # ` W ) - 2 ) + 2 ) = ( # ` W ) )
57	56	eqcomd 2475	. . . . . 6 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
58	2, 55, 57	3syl 20	. . . . 5 |- ( W e. Word V -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
59	58	adantr 465	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
60	59	opeq2d 4225	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <. ( ( # ` W ) - 2 ) , ( # ` W ) >. = <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. )
61	60	oveq2d 6310	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) )
62		eqidd 2468	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W ` ( ( # ` W ) - 2 ) ) = ( W ` ( ( # ` W ) - 2 ) ) )
63		lsw 12560	. . . . 5 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
64	40, 44	syl 16	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
65	64	eqcomd 2475	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
66		2m1e1 10660	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
67	66	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 2 - 1 ) = 1 )
68	67	oveq2d 6310	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( # ` W ) - 1 ) )
69	65, 68	eqtrd 2508	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
70	2, 69	syl 16	. . . . . . 7 |- ( W e. Word V -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
71	70	eqcomd 2475	. . . . . 6 |- ( W e. Word V -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
72	71	fveq2d 5875	. . . . 5 |- ( W e. Word V -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
73	63, 72	eqtrd 2508	. . . 4 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
74	73	adantr 465	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
75	62, 74	s2eqd 12802	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
76	54, 61, 75	3eqtr4d 2518	1 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   <.cop 4038   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   CCcc 9500   RRcr 9501   0cc0 9502   1c1 9503   + caddc 9505   < clt 9638   <_ cle 9639   - cmin 9815   NNcn 10546   2c2 10595   NN0cn0 10805   ZZcz 10874  ..^cfzo 11802   #chash 12383  Word cword 12510   lastS clsw 12511   substr csubstr 12514   <"cs2 12781

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-oadd 7144  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-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-hash 12384  df-word 12518  df-lsw 12519  df-concat 12520  df-s1 12521  df-substr 12522  df-s2 12788

This theorem is referenced by:  2swrd2eqwrdeq  12866