Theorem swrd2lsw 13695

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 473	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> W e. Word V )
2		lencl 13324	. . . . 5 |- ( W e. Word V -> ( # ` W ) e. NN0 )
3		1z 11407	. . . . . . . . 9 |- 1 e. ZZ
4		nn0z 11400	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
5		zltp1le 11427	. . . . . . . . 9 |- ( ( 1 e. ZZ /\ ( # ` W ) e. ZZ ) -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
6	3, 4, 5	sylancr 695	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
7		1p1e2 11134	. . . . . . . . . . 11 |- ( 1 + 1 ) = 2
8	7	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 1 + 1 ) = 2 )
9	8	breq1d 4663	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) <-> 2 <_ ( # ` W ) ) )
10	9	biimpd 219	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) -> 2 <_ ( # ` W ) ) )
11	6, 10	sylbid 230	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> 2 <_ ( # ` W ) ) )
12	11	imp 445	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> 2 <_ ( # ` W ) )
13		2nn0 11309	. . . . . . . . 9 |- 2 e. NN0
14	13	jctl 564	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) )
15	14	adantr 481	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) )
16		nn0sub 11343	. . . . . . 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 222	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
19	2, 18	sylan 488	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
20		0red 10041	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 0 e. RR )
21		1red 10055	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> 1 e. RR )
22		zre 11381	. . . . . . . . . . . 12 |- ( ( # ` W ) e. ZZ -> ( # ` W ) e. RR )
23	20, 21, 22	3jca 1242	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) )
24		0lt1 10550	. . . . . . . . . . 11 |- 0 < 1
25		lttr 10114	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( # ` W ) ) -> 0 < ( # ` W ) ) )
26	25	expd 452	. . . . . . . . . . 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 11387	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. ZZ /\ 0 < ( # ` W ) ) )
29	28	simplbi2 655	. . . . . . . . . 10 |- ( ( # ` W ) e. ZZ -> ( 0 < ( # ` W ) -> ( # ` W ) e. NN ) )
30	27, 29	syld 47	. . . . . . . . 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 445	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( # ` W ) e. NN )
33		fzo0end 12560	. . . . . . 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 11302	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. CC )
36		2cn 11091	. . . . . . . . . . . 12 |- 2 e. CC
37	36	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 2 e. CC )
38		1cnd 10056	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> 1 e. CC )
39	35, 37, 38	3jca 1242	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) )
40		1e2m1 11136	. . . . . . . . . . . . 13 |- 1 = ( 2 - 1 )
41	40	a1i 11	. . . . . . . . . . . 12 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> 1 = ( 2 - 1 ) )
42	41	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
43		subsub 10311	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 2 e. CC /\ 1 e. CC ) -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( ( # ` W ) - 2 ) + 1 ) )
44	42, 43	eqtrd 2656	. . . . . . . . . 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 2628	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - 1 ) )
47	46	eleq1d 2686	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
48	47	adantr 481	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) <-> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) )
49	34, 48	mpbird 247	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
50	2, 49	sylan 488	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) )
51	1, 19, 50	3jca 1242	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W e. Word V /\ ( ( # ` W ) - 2 ) e. NN0 /\ ( ( ( # ` W ) - 2 ) + 1 ) e. ( 0..^ ( # ` W ) ) ) )
52		swrds2 13685	. . 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 561	. . . . . 6 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) e. CC /\ 2 e. CC ) )
55		npcan 10290	. . . . . . 7 |- ( ( ( # ` W ) e. CC /\ 2 e. CC ) -> ( ( ( # ` W ) - 2 ) + 2 ) = ( # ` W ) )
56	55	eqcomd 2628	. . . . . 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 481	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( # ` W ) = ( ( ( # ` W ) - 2 ) + 2 ) )
59	58	opeq2d 4409	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <. ( ( # ` W ) - 2 ) , ( # ` W ) >. = <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. )
60	59	oveq2d 6666	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( W substr <. ( ( # ` W ) - 2 ) , ( ( ( # ` W ) - 2 ) + 2 ) >. ) )
61		eqidd 2623	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W ` ( ( # ` W ) - 2 ) ) = ( W ` ( ( # ` W ) - 2 ) ) )
62		lsw 13351	. . . . 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 2628	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( ( # ` W ) - 2 ) + 1 ) = ( ( # ` W ) - ( 2 - 1 ) ) )
65		2m1e1 11135	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
66	65	a1i 11	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( 2 - 1 ) = 1 )
67	66	oveq2d 6666	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - ( 2 - 1 ) ) = ( ( # ` W ) - 1 ) )
68	64, 67	eqtrd 2656	. . . . . . . 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 2628	. . . . . 6 |- ( W e. Word V -> ( ( # ` W ) - 1 ) = ( ( ( # ` W ) - 2 ) + 1 ) )
71	70	fveq2d 6195	. . . . 5 |- ( W e. Word V -> ( W ` ( ( # ` W ) - 1 ) ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
72	62, 71	eqtrd 2656	. . . 4 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
73	72	adantr 481	. . 3 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( lastS ` W ) = ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) )
74	61, 73	s2eqd 13608	. 2 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( W ` ( ( # ` W ) - 2 ) ) ( W ` ( ( ( # ` W ) - 2 ) + 1 ) ) "> )
75	53, 60, 74	3eqtr4d 2666	1 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295   <"cs2 13586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-s2 13593

This theorem is referenced by:  2swrd2eqwrdeq  13696