Theorem swrd2lsw 12869

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 12541	. . . . 5 |- ( W e. Word V -> ( # ` W ) e. NN0 )
3		1z 10900	. . . . . . . . 9 |- 1 e. ZZ
4		nn0z 10893	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
5		zltp1le 10919	. . . . . . . . 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 10655	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
8	7	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 1 + 1 ) = 2 )
9	8	breq1d 4447	. . . . . . . . 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 10818	. . . . . . . . 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 10852	. . . . . . 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 9600	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 0 e. RR )
21		1red 9614	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 1 e. RR )
22		zre 10874	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> ( # ` W ) e. RR )
23	20, 21, 22	3jca 1177	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) )
24		0lt1 10081	. . . . . . . . . . 11 |- 0 < 1
25		lttr 9664	. . . . . . . . . . . 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 10880	. . . . . . . . . . 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 11883	. . . . . . 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 10811	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
36		2cn 10612	. . . . . . . . . . . 12 |- 2 e. CC
37	36	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 2 e. CC )
38		1cnd 9615	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 1 e. CC )
39	35, 37, 38	3jca 1177	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) )
40		1e2m1 10657	. . . . . . . . . . . . 13 |- 1 = ( 2 - 1 )
41	40	a1i 11	. . . . . . . . . . . 12 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> 1 = ( 2 - 1 ) )
42	41	oveq2d 6297	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
43		subsub 9854	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
44	42, 43	eqtrd 2484	. . . . . . . . . 10 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
45	39, 44	syl 16	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
46	45	eqcomd 2451	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
47	46	eleq1d 2512	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
48	47	adantr 465	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
49	34, 48	mpbird 232	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
50	2, 49	sylan 471	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
51	1, 19, 50	3jca 1177	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W e. Word V /\ ( ( # ` W ) - 2 ) e. NN0 /\ ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) ) )
52		swrds2 12862	. . 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 16	. 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 538	. . . . . 6 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC ) )
55		npcan 9834	. . . . . . 7 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( ( ( # ` W ) - 2 ) + 2 ) = ( # ` W ) )
56	55	eqcomd 2451	. . . . . 6 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
57	2, 54, 56	3syl 20	. . . . 5 |- ( W e. Word V -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
58	57	adantr 465	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
59	58	opeq2d 4209	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <. ( ( # ` W ) - 2 ) , ( # ` W ) >. = <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. )
60	59	oveq2d 6297	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) )
61		eqidd 2444	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W ` ( ( # ` W ) - 2 ) ) = ( W ` ( ( # ` W ) - 2 ) ) )
62		lsw 12564	. . . . 5 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
63	39, 43	syl 16	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
64	63	eqcomd 2451	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
65		2m1e1 10656	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
66	65	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 2 - 1 ) = 1 )
67	66	oveq2d 6297	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( # ` W ) - 1 ) )
68	64, 67	eqtrd 2484	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
69	2, 68	syl 16	. . . . . . 7 |- ( W e. Word V -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
70	69	eqcomd 2451	. . . . . 6 |- ( W e. Word V -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
71	70	fveq2d 5860	. . . . 5 |- ( W e. Word V -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
72	62, 71	eqtrd 2484	. . . 4 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
73	72	adantr 465	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
74	61, 73	s2eqd 12806	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
75	53, 60, 74	3eqtr4d 2494	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 974   = wceq 1383   e. wcel 1804   <.cop 4020   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   + caddc 9498   < clt 9631   <_ cle 9632   - cmin 9810   NNcn 10542   2c2 10591   NN0cn0 10801   ZZcz 10870  ..^cfzo 11803   #chash 12384  Word cword 12513   lastS clsw 12514   substr csubstr 12517   <"cs2 12785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-hash 12385  df-word 12521  df-lsw 12522  df-concat 12523  df-s1 12524  df-substr 12525  df-s2 12792

This theorem is referenced by:  2swrd2eqwrdeq  12870