Theorem swrd2lsw 13075

Description: Extract the last two symbols 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 463	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> W e. Word V )
2		lencl 12721	. . . . 5 |- ( W e. Word V -> ( # ` W ) e. NN0 )
3		1z 10995	. . . . . . . . 9 |- 1 e. ZZ
4		nn0z 10988	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
5		zltp1le 11014	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ ( # ` W ) e. ZZ ) -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
6	3, 4, 5	sylancr 674	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
7		1p1e2 10750	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
8	7	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 1 + 1 ) = 2 )
9	8	breq1d 4425	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) <-> 2 <_ ( # ` W ) ) )
10	9	biimpd 212	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) -> 2 <_ ( # ` W ) ) )
11	6, 10	sylbid 223	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> 2 <_ ( # ` W ) ) )
12	11	imp 435	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> 2 <_ ( # ` W ) )
13		2nn0 10914	. . . . . . . . 9 |- 2 e. NN0
14	13	jctl 548	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) )
15	14	adantr 471	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) )
16		nn0sub 10948	. . . . . . 7 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) -> ( 2 <_ ( # ` W ) <-> ( ( # ` W ) - 2 ) e. NN0 ) )
17	15, 16	syl 17	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( 2 <_ ( # ` W ) <-> ( ( # ` W ) - 2 ) e. NN0 ) )
18	12, 17	mpbid 215	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
19	2, 18	sylan 478	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
20		0red 9669	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 0 e. RR )
21		1red 9683	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 1 e. RR )
22		zre 10969	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> ( # ` W ) e. RR )
23	20, 21, 22	3jca 1194	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) )
24		0lt1 10163	. . . . . . . . . . 11 |- 0 < 1
25		lttr 9735	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( # ` W ) ) -> 0 < ( # ` W ) ) )
26	25	expd 442	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( 0 < 1 -> ( 1 < ( # ` W ) -> 0 < ( # ` W ) ) ) )
27	23, 24, 26	mpisyl 21	. . . . . . . . . 10 |- ( ( # ` W ) e. ZZ -> ( 1 < ( # ` W ) -> 0 < ( # ` W ) ) )
28		elnnz 10975	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. ZZ /\ 0 < ( # ` W ) ) )
29	28	simplbi2 635	. . . . . . . . . 10 |- ( ( # ` W ) e. ZZ -> ( 0 < ( # ` W ) -> ( # ` W ) e. NN ) )
30	27, 29	syld 45	. . . . . . . . 9 |- ( ( # ` W ) e. ZZ -> ( 1 < ( # ` W ) -> ( # ` W ) e. NN ) )
31	4, 30	syl 17	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> ( # ` W ) e. NN ) )
32	31	imp 435	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( # ` W ) e. NN )
33		fzo0end 12033	. . . . . . 7 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
34	32, 33	syl 17	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
35		nn0cn 10907	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
36		2cn 10707	. . . . . . . . . . . 12 |- 2 e. CC
37	36	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 2 e. CC )
38		1cnd 9684	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 1 e. CC )
39	35, 37, 38	3jca 1194	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) )
40		1e2m1 10752	. . . . . . . . . . . . 13 |- 1 = ( 2 - 1 )
41	40	a1i 11	. . . . . . . . . . . 12 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> 1 = ( 2 - 1 ) )
42	41	oveq2d 6330	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
43		subsub 9929	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
44	42, 43	eqtrd 2495	. . . . . . . . . 10 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
45	39, 44	syl 17	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
46	45	eqcomd 2467	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
47	46	eleq1d 2523	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
48	47	adantr 471	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
49	34, 48	mpbird 240	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
50	2, 49	sylan 478	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
51	1, 19, 50	3jca 1194	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W e. Word V /\ ( ( # ` W ) - 2 ) e. NN0 /\ ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) ) )
52		swrds2 13065	. . 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 ) ) "> )
53	51, 52	syl 17	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
54	35, 36	jctir 545	. . . . . 6 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC ) )
55		npcan 9909	. . . . . . 7 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( ( ( # ` W ) - 2 ) + 2 ) = ( # ` W ) )
56	55	eqcomd 2467	. . . . . 6 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
57	2, 54, 56	3syl 18	. . . . 5 |- ( W e. Word V -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
58	57	adantr 471	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
59	58	opeq2d 4186	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <. ( ( # ` W ) - 2 ) , ( # ` W ) >. = <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. )
60	59	oveq2d 6330	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) )
61		eqidd 2462	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W ` ( ( # ` W ) - 2 ) ) = ( W ` ( ( # ` W ) - 2 ) ) )
62		lsw 12746	. . . . 5 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
63	39, 43	syl 17	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
64	63	eqcomd 2467	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
65		2m1e1 10751	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
66	65	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 2 - 1 ) = 1 )
67	66	oveq2d 6330	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( # ` W ) - 1 ) )
68	64, 67	eqtrd 2495	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
69	2, 68	syl 17	. . . . . . 7 |- ( W e. Word V -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
70	69	eqcomd 2467	. . . . . 6 |- ( W e. Word V -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
71	70	fveq2d 5891	. . . . 5 |- ( W e. Word V -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
72	62, 71	eqtrd 2495	. . . 4 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
73	72	adantr 471	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
74	61, 73	s2eqd 12994	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
75	53, 60, 74	3eqtr4d 2505	1 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   <.cop 3985   class class class wbr 4415   ` cfv 5600  (class class class)co 6314   CCcc 9562   RRcr 9563   0cc0 9564   1c1 9565   + caddc 9567   < clt 9700   <_ cle 9701   - cmin 9885   NNcn 10636   2c2 10686   NN0cn0 10897   ZZcz 10965  ..^cfzo 11945   #chash 12546  Word cword 12688   lastS clsw 12689   substr csubstr 12692   <"cs2 12973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-hash 12547  df-word 12696  df-lsw 12697  df-concat 12698  df-s1 12699  df-substr 12700  df-s2 12980

This theorem is referenced by:  2swrd2eqwrdeq  13076