Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlkextinj Structured version   Visualization version   Unicode version

Theorem wwlkextinj 25470
 Description: Lemma 2 for wwlkextbij 25473. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
Ref Expression
wwlkextbij.d Word substr lastS lastS
wwlkextbij.r lastS
wwlkextbij.f lastS
Assertion
Ref Expression
wwlkextinj
Distinct variable groups:   ,   ,,   ,,   ,   ,,,   ,,,
Allowed substitution hints:   (,)   (,)   ()   (,,)   ()

Proof of Theorem wwlkextinj
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlkextbij.d . . 3 Word substr lastS lastS
2 wwlkextbij.r . . 3 lastS
3 wwlkextbij.f . . 3 lastS
41, 2, 3wwlkextfun 25469 . 2
5 fveq2 5870 . . . . . . 7 lastS lastS
6 fvex 5880 . . . . . . 7 lastS
75, 3, 6fvmpt 5953 . . . . . 6 lastS
8 fveq2 5870 . . . . . . 7 lastS lastS
9 fvex 5880 . . . . . . 7 lastS
108, 3, 9fvmpt 5953 . . . . . 6 lastS
117, 10eqeqan12d 2469 . . . . 5 lastS lastS
1211adantl 468 . . . 4 lastS lastS
13 fveq2 5870 . . . . . . . . 9
1413eqeq1d 2455 . . . . . . . 8
15 oveq1 6302 . . . . . . . . 9 substr substr
1615eqeq1d 2455 . . . . . . . 8 substr substr
17 fveq2 5870 . . . . . . . . . 10 lastS lastS
1817preq2d 4061 . . . . . . . . 9 lastS lastS lastS lastS
1918eleq1d 2515 . . . . . . . 8 lastS lastS lastS lastS
2014, 16, 193anbi123d 1341 . . . . . . 7 substr lastS lastS substr lastS lastS
2120, 1elrab2 3200 . . . . . 6 Word substr lastS lastS
22 fveq2 5870 . . . . . . . . 9
2322eqeq1d 2455 . . . . . . . 8
24 oveq1 6302 . . . . . . . . 9 substr substr
2524eqeq1d 2455 . . . . . . . 8 substr substr
26 fveq2 5870 . . . . . . . . . 10 lastS lastS
2726preq2d 4061 . . . . . . . . 9 lastS lastS lastS lastS
2827eleq1d 2515 . . . . . . . 8 lastS lastS lastS lastS
2923, 25, 283anbi123d 1341 . . . . . . 7 substr lastS lastS substr lastS lastS
3029, 1elrab2 3200 . . . . . 6 Word substr lastS lastS
31 eqtr3 2474 . . . . . . . . . . . . . . . . 17
3231expcom 437 . . . . . . . . . . . . . . . 16
33323ad2ant1 1030 . . . . . . . . . . . . . . 15 substr lastS lastS
3433adantl 468 . . . . . . . . . . . . . 14 Word substr lastS lastS
3534com12 32 . . . . . . . . . . . . 13 Word substr lastS lastS
36353ad2ant1 1030 . . . . . . . . . . . 12 substr lastS lastS Word substr lastS lastS
3736adantl 468 . . . . . . . . . . 11 Word substr lastS lastS Word substr lastS lastS
3837imp 431 . . . . . . . . . 10 Word substr lastS lastS Word substr lastS lastS
3938adantr 467 . . . . . . . . 9 Word substr lastS lastS Word substr lastS lastS
4039adantr 467 . . . . . . . 8 Word substr lastS lastS Word substr lastS lastS lastS lastS
41 simpr 463 . . . . . . . 8 Word substr lastS lastS Word substr lastS lastS lastS lastS lastS lastS
42 eqtr3 2474 . . . . . . . . . . . . . . . . . . . 20 substr substr substr substr
43 1e2m1 10732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4544oveq2d 6311 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
46 nn0cn 10886 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
47 2cnd 10689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
48 1cnd 9664 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4946, 47, 48addsubassd 10011 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5045, 49eqtr4d 2490 . . . . . . . . . . . . . . . . . . . . . . . . . 26
5150adantr 467 . . . . . . . . . . . . . . . . . . . . . . . . 25
52 oveq1 6302 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5352eqeq2d 2463 . . . . . . . . . . . . . . . . . . . . . . . . . 26
5453adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . 25
5551, 54mpbird 236 . . . . . . . . . . . . . . . . . . . . . . . 24
56 opeq2 4170 . . . . . . . . . . . . . . . . . . . . . . . . . 26
5756oveq2d 6311 . . . . . . . . . . . . . . . . . . . . . . . . 25 substr substr
5856oveq2d 6311 . . . . . . . . . . . . . . . . . . . . . . . . 25 substr substr
5957, 58eqeq12d 2468 . . . . . . . . . . . . . . . . . . . . . . . 24 substr substr substr substr
6055, 59syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 substr substr substr substr
6160biimpd 211 . . . . . . . . . . . . . . . . . . . . . 22 substr substr substr substr
6261ex 436 . . . . . . . . . . . . . . . . . . . . 21 substr substr substr substr
6362com13 83 . . . . . . . . . . . . . . . . . . . 20 substr substr substr substr
6442, 63syl 17 . . . . . . . . . . . . . . . . . . 19 substr substr substr substr
6564ex 436 . . . . . . . . . . . . . . . . . 18 substr substr substr substr
6665com23 81 . . . . . . . . . . . . . . . . 17 substr substr substr substr
6766impcom 432 . . . . . . . . . . . . . . . 16 substr substr substr substr
6867com12 32 . . . . . . . . . . . . . . 15 substr substr substr substr
69683ad2ant2 1031 . . . . . . . . . . . . . 14 substr lastS lastS substr substr substr
7069adantl 468 . . . . . . . . . . . . 13 Word substr lastS lastS substr substr substr
7170com12 32 . . . . . . . . . . . 12 substr Word substr lastS lastS substr substr
72713adant3 1029 . . . . . . . . . . 11 substr lastS lastS Word substr lastS lastS substr substr
7372adantl 468 . . . . . . . . . 10 Word substr lastS lastS Word substr lastS lastS substr substr
7473imp31 434 . . . . . . . . 9 Word substr lastS lastS Word substr lastS lastS substr substr
7574adantr 467 . . . . . . . 8 Word substr lastS lastS Word substr lastS lastS lastS lastS substr substr
76 simpl 459 . . . . . . . . . . . . 13 Word substr lastS lastS Word
77 simpl 459 . . . . . . . . . . . . 13 Word substr lastS lastS Word
7876, 77anim12i 570 . . . . . . . . . . . 12 Word substr lastS lastS Word substr lastS lastS Word Word
7978adantr 467 . . . . . . . . . . 11 Word substr lastS lastS Word substr lastS lastS Word Word
80 nn0re 10885 . . . . . . . . . . . . . . . . . . . . 21
81 2re 10686 . . . . . . . . . . . . . . . . . . . . . 22
8281a1i 11 . . . . . . . . . . . . . . . . . . . . 21
83 nn0ge0 10902 . . . . . . . . . . . . . . . . . . . . 21
84 2pos 10708 . . . . . . . . . . . . . . . . . . . . . 22
8584a1i 11 . . . . . . . . . . . . . . . . . . . . 21
8680, 82, 83, 85addgegt0d 10194 . . . . . . . . . . . . . . . . . . . 20
8786adantl 468 . . . . . . . . . . . . . . . . . . 19
88 breq2 4409 . . . . . . . . . . . . . . . . . . . 20
8988adantr 467 . . . . . . . . . . . . . . . . . . 19
9087, 89mpbird 236 . . . . . . . . . . . . . . . . . 18
91 hashgt0n0 12553 . . . . . . . . . . . . . . . . . 18 Word
9290, 91sylan2 477 . . . . . . . . . . . . . . . . 17 Word
9392exp32 610 . . . . . . . . . . . . . . . 16 Word
9493com12 32 . . . . . . . . . . . . . . 15 Word
95943ad2ant1 1030 . . . . . . . . . . . . . 14 substr lastS lastS Word
9695impcom 432 . . . . . . . . . . . . 13 Word substr lastS lastS
9796adantr 467 . . . . . . . . . . . 12 Word substr lastS lastS Word substr lastS lastS
9897imp 431 . . . . . . . . . . 11 Word substr lastS lastS Word substr lastS lastS
9986adantl 468 . . . . . . . . . . . . . . . . . . 19
100 breq2 4409 . . . . . . . . . . . . . . . . . . . 20
101100adantr 467 . . . . . . . . . . . . . . . . . . 19
10299, 101mpbird 236 . . . . . . . . . . . . . . . . . 18
103 hashgt0n0 12553 . . . . . . . . . . . . . . . . . 18 Word
104102, 103sylan2 477 . . . . . . . . . . . . . . . . 17 Word
105104exp32 610 . . . . . . . . . . . . . . . 16 Word
106105com12 32 . . . . . . . . . . . . . . 15 Word
1071063ad2ant1 1030 . . . . . . . . . . . . . 14 substr lastS lastS Word
108107impcom 432 . . . . . . . . . . . . 13 Word substr lastS lastS
109108adantl 468 . . . . . . . . . . . 12 Word substr lastS lastS Word substr lastS lastS
110109imp 431 . . . . . . . . . . 11 Word substr lastS lastS Word substr lastS lastS
11179, 98, 110jca32 538 . . . . . . . . . 10 Word substr lastS lastS Word substr lastS lastS Word Word
112111adantr 467 . . . . . . . . 9 Word substr lastS lastS Word substr lastS lastS lastS lastS Word Word
113 simpl 459 . . . . . . . . . . . 12 Word Word Word
114113adantr 467 . . . . . . . . . . 11 Word Word Word
115 simpr 463 . . . . . . . . . . . 12 Word Word Word
116115adantr 467 . . . . . . . . . . 11 Word Word Word
117 hashneq0 12552 . . . . . . . . . . . . . . . 16 Word
118117biimprd 227 . . . . . . . . . . . . . . 15 Word
119118adantr 467 . . . . . . . . . . . . . 14 Word Word
120119com12 32 . . . . . . . . . . . . 13 Word Word
121120adantr 467 . . . . . . . . . . . 12 Word Word
122121impcom 432 . . . . . . . . . . 11 Word Word
123 2swrd1eqwrdeq 12817 . . . . . . . . . . 11 Word Word substr substr lastS lastS
124114, 116, 122, 123syl3anc 1269 . . . . . . . . . 10 Word Word substr substr lastS lastS
125 ancom 452 . . . . . . . . . . . 12 substr substr lastS lastS lastS lastS substr substr
126125anbi2i 701 . . . . . . . . . . 11 substr substr lastS lastS lastS lastS substr substr
127 3anass 990 . . . . . . . . . . 11 lastS lastS substr substr lastS lastS substr substr
128126, 127bitr4i 256 . . . . . . . . . 10 substr substr lastS lastS lastS lastS substr substr
129124, 128syl6bb 265 . . . . . . . . 9 Word Word lastS lastS substr substr
130112, 129syl 17 . . . . . . . 8 Word substr lastS lastS Word substr lastS lastS lastS lastS lastS lastS substr substr
13140, 41, 75, 130mpbir3and 1192 . . . . . . 7 Word substr lastS lastS Word substr lastS lastS lastS lastS
132131exp31 609 . . . . . 6 Word substr lastS lastS Word substr lastS lastS lastS lastS
13321, 30, 132syl2anb 482 . . . . 5 lastS lastS
134133impcom 432 . . . 4 lastS lastS
13512, 134sylbid 219 . . 3
136135ralrimivva 2811 . 2
137 dff13 6164 . 2
1384, 136, 137sylanbrc 671 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   w3a 986   wceq 1446   wcel 1889   wne 2624  wral 2739  crab 2743  c0 3733  cpr 3972  cop 3976   class class class wbr 4405   cmpt 4464   crn 4838  wf 5581  wf1 5582  cfv 5585  (class class class)co 6295  cr 9543  cc0 9544  c1 9545   caddc 9547   clt 9680   cmin 9865  c2 10666  cn0 10876  chash 12522  Word cword 12663   lastS clsw 12664   substr csubstr 12667 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-lsw 12672  df-s1 12674  df-substr 12675 This theorem is referenced by:  wwlkextbij0  25472
 Copyright terms: Public domain W3C validator