Theorem swrd2lsw 12552

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 12249	. . . . 5 |- ( W e. Word V -> ( # ` W ) e. NN0 )
3		1z 10676	. . . . . . . . 9 |- 1 e. ZZ
4		nn0z 10669	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
5		zltp1le 10694	. . . . . . . . 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 10435	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
8	7	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 1 + 1 ) = 2 )
9	8	breq1d 4302	. . . . . . . . 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 10596	. . . . . . . . 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 10630	. . . . . . 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 9387	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 0 e. RR )
21		1red 9401	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 1 e. RR )
22		zre 10650	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> ( # ` W ) e. RR )
23	20, 21, 22	3jca 1168	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) )
24		0lt1 9862	. . . . . . . . . . 11 |- 0 < 1
25		lttr 9451	. . . . . . . . . . . 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 10656	. . . . . . . . . . 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 11619	. . . . . . 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 10589	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
36		2cn 10392	. . . . . . . . . . . 12 |- 2 e. CC
37	36	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 2 e. CC )
38		ax-1cn 9340	. . . . . . . . . . . 12 |- 1 e. CC
39	38	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 1 e. CC )
40	35, 37, 39	3jca 1168	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) )
41		1e2m1 10437	. . . . . . . . . . . . 13 |- 1 = ( 2 - 1 )
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> 1 = ( 2 - 1 ) )
43	42	oveq2d 6107	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
44		subsub 9639	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
45	43, 44	eqtrd 2475	. . . . . . . . . 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 2448	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
48	47	eleq1d 2509	. . . . . . 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 1168	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W e. Word V /\ ( ( # ` W ) - 2 ) e. NN0 /\ ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) ) )
53		swrds2 12545	. . 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 9619	. . . . . . 7 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( ( ( # ` W ) - 2 ) + 2 ) = ( # ` W ) )
57	56	eqcomd 2448	. . . . . 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 4066	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <. ( ( # ` W ) - 2 ) , ( # ` W ) >. = <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. )
61	60	oveq2d 6107	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) )
62		eqidd 2444	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W ` ( ( # ` W ) - 2 ) ) = ( W ` ( ( # ` W ) - 2 ) ) )
63		lsw 12266	. . . . 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 2448	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
66		2m1e1 10436	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
67	66	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 2 - 1 ) = 1 )
68	67	oveq2d 6107	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( # ` W ) - 1 ) )
69	65, 68	eqtrd 2475	. . . . . . . 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 2448	. . . . . 6 |- ( W e. Word V -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
72	71	fveq2d 5695	. . . . 5 |- ( W e. Word V -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
73	63, 72	eqtrd 2475	. . . 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 12489	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
76	54, 61, 75	3eqtr4d 2485	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 965   = wceq 1369   e. wcel 1756   <.cop 3883   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283   + caddc 9285   < clt 9418   <_ cle 9419   - cmin 9595   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646  ..^cfzo 11548   #chash 12103  Word cword 12221   lastS clsw 12222   substr csubstr 12225   <"cs2 12468

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-oadd 6924  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-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-lsw 12230  df-concat 12231  df-s1 12232  df-substr 12233  df-s2 12475

This theorem is referenced by:  2swrd2eqwrdeq  12553