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| Mirrors > Home > MPE Home > Th. List > pthdlem1 | Structured version Visualization version Unicode version | ||
| Description: Lemma 1 for pthd 26665. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.) |
| Ref | Expression |
|---|---|
| pthd.p |
      Word   |
| pthd.r |
      |
| pthd.s |
   ..^ ..^  |
| Ref | Expression |
|---|---|
| pthdlem1 |
   ..^ |
,, ,, ,,| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pthd.p |
. . . . . . . 8
Word | |
| 2 | wrdf 13310 |
. . . . . . . 8
Word
 ..^ | |
| 3 | 1, 2 | syl 17 |
. . . . . . 7
..^ |
| 4 | fzo0ss1 12498 |
. . . . . . . . 9
..^  ..^ | |
| 5 | pthd.r |
. . . . . . . . . . 11
| |
| 6 | 5 | a1i 11 |
. . . . . . . . . 10
|
| 7 | 6 | oveq2d 6666 |
. . . . . . . . 9
..^ ..^ |
| 8 | 4, 7 | syl5sseq 3653 |
. . . . . . . 8
..^ ..^ |
| 9 | lencl 13324 |
. . . . . . . . . 10
Word
 | |
| 10 | nn0z 11400 |
. . . . . . . . . 10
 | |
| 11 | 1, 9, 10 | 3syl 18 |
. . . . . . . . 9
|
| 12 | fzossrbm1 12497 |
. . . . . . . . 9
..^
..^ | |
| 13 | 11, 12 | syl 17 |
. . . . . . . 8
..^ ..^ |
| 14 | 8, 13 | sstrd 3613 |
. . . . . . 7
..^ ..^ |
| 15 | 3, 14 | fssresd 6071 |
. . . . . 6
..^..^ |
| 16 | 15 | adantr 481 |
. . . . 5
![/\](wedge.gif "/\"){kind=small}
..^..^ |
| 17 | pthd.s |
. . . . . . 7
..^ ..^ | |
| 18 | 17 | adantr 481 |
. . . . . 6
..^
..^ |
| 19 | 1, 9 | syl 17 |
. . . . . . . . . . . 12
|
| 20 | nn0re 11301 |
. . . . . . . . . . . . . . . 16
 | |
| 21 | 20 | ltm1d 10956 |
. . . . . . . . . . . . . . 15
|
| 22 | 1re 10039 |
. . . . . . . . . . . . . . . . 17
| |
| 23 | peano2rem 10348 |
. . . . . . . . . . . . . . . . . 18
| |
| 24 | 20, 23 | syl 17 |
. . . . . . . . . . . . . . . . 17
|
| 25 | lttr 10114 |
. . . . . . . . . . . . . . . . 17
| |
| 26 | 22, 24, 20, 25 | mp3an2i 1429 |
. . . . . . . . . . . . . . . 16
|
| 27 | 1red 10055 |
. . . . . . . . . . . . . . . . 17
| |
| 28 | ltle 10126 |
. . . . . . . . . . . . . . . . 17
| |
| 29 | 27, 20, 28 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
|
| 30 | 26, 29 | syld 47 |
. . . . . . . . . . . . . . 15
|
| 31 | 21, 30 | mpan2d 710 |
. . . . . . . . . . . . . 14
|
| 32 | 31 | imdistani 726 |
. . . . . . . . . . . . 13
|
| 33 | elnnnn0c 11338 |
. . . . . . . . . . . . 13
 
| |
| 34 | 32, 33 | sylibr 224 |
. . . . . . . . . . . 12
|
| 35 | 19, 34 | sylan 488 |
. . . . . . . . . . 11
|
| 36 | fzo0sn0fzo1 12557 |
. . . . . . . . . . 11
..^    ..^ | |
| 37 | 35, 36 | syl 17 |
. . . . . . . . . 10
..^ ..^ |
| 38 | 1zzd 11408 |
. . . . . . . . . . . 12
| |
| 39 | 1p1e2 11134 |
. . . . . . . . . . . . . . . 16
  | |
| 40 | 2z 11409 |
. . . . . . . . . . . . . . . 16
| |
| 41 | 39, 40 | eqeltri 2697 |
. . . . . . . . . . . . . . 15
|
| 42 | 41 | a1i 11 |
. . . . . . . . . . . . . 14
|
| 43 | 10 | adantr 481 |
. . . . . . . . . . . . . 14
|
| 44 | ltaddsub 10502 |
. . . . . . . . . . . . . . . . . 18
| |
| 45 | 44 | bicomd 213 |
. . . . . . . . . . . . . . . . 17
|
| 46 | 22, 27, 20, 45 | mp3an2i 1429 |
. . . . . . . . . . . . . . . 16
|
| 47 | 2re 11090 |
. . . . . . . . . . . . . . . . . 18
| |
| 48 | 39, 47 | eqeltri 2697 |
. . . . . . . . . . . . . . . . 17
|
| 49 | ltle 10126 |
. . . . . . . . . . . . . . . . 17
| |
| 50 | 48, 20, 49 | sylancr 695 |
. . . . . . . . . . . . . . . 16
|
| 51 | 46, 50 | sylbid 230 |
. . . . . . . . . . . . . . 15
|
| 52 | 51 | imp 445 |
. . . . . . . . . . . . . 14
|
| 53 | eluz2 11693 |
. . . . . . . . . . . . . 14
| |
| 54 | 42, 43, 52, 53 | syl3anbrc 1246 |
. . . . . . . . . . . . 13
|
| 55 | 19, 54 | sylan 488 |
. . . . . . . . . . . 12
|
| 56 | fzosplitsnm1 12542 |
. . . . . . . . . . . 12
..^ ..^ | |
| 57 | 38, 55, 56 | syl2anc 693 |
. . . . . . . . . . 11
..^ ..^ |
| 58 | 57 | uneq2d 3767 |
. . . . . . . . . 10
..^ ..^ |
| 59 | 37, 58 | eqtrd 2656 |
. . . . . . . . 9
..^ ..^ |
| 60 | 59 | raleqdv 3144 |
. . . . . . . 8
..^ ..^
..^
..^ |
| 61 | ralunb 3794 |
. . . . . . . . 9
..^ ..^
..^
..^ ..^ | |
| 62 | ralunb 3794 |
. . . . . . . . . 10
..^ ..^ ..^ ..^ ..^ | |
| 63 | 62 | anbi2i 730 |
. . . . . . . . 9
..^ ..^ ..^
..^
..^ ..^ ..^ |
| 64 | 61, 63 | bitri 264 |
. . . . . . . 8
..^ ..^
..^
..^ ..^ ..^ |
| 65 | 60, 64 | syl6bb 276 |
. . . . . . 7
..^ ..^
..^
..^ ..^ ..^ |
| 66 | 5 | eqcomi 2631 |
. . . . . . . . . . . 12
|
| 67 | 66 | oveq2i 6661 |
. . . . . . . . . . 11
..^ ..^ |
| 68 | 67 | raleqi 3142 |
. . . . . . . . . 10
..^ ..^
..^
..^ |
| 69 | fvres 6207 |
. . . . . . . . . . . . . . . . . 18
..^
..^ | |
| 70 | 69 | eqcomd 2628 |
. . . . . . . . . . . . . . . . 17
..^
..^ |
| 71 | 70 | adantl 482 |
. . . . . . . . . . . . . . . 16
..^
..^ |
| 72 | 71 | adantr 481 |
. . . . . . . . . . . . . . 15
..^ ..^
..^ |
| 73 | fvres 6207 |
. . . . . . . . . . . . . . . . 17
..^
..^ | |
| 74 | 73 | eqcomd 2628 |
. . . . . . . . . . . . . . . 16
..^
..^ |
| 75 | 74 | adantl 482 |
. . . . . . . . . . . . . . 15
..^ ..^
..^ |
| 76 | 72, 75 | neeq12d 2855 |
. . . . . . . . . . . . . 14
..^ ..^
..^ ..^ |
| 77 | 76 | biimpd 219 |
. . . . . . . . . . . . 13
..^ ..^
..^
..^ |
| 78 | 77 | imim2d 57 |
. . . . . . . . . . . 12
..^ ..^
..^
..^ |
| 79 | 78 | ralimdva 2962 |
. . . . . . . . . . 11
..^
..^ ..^ ..^ ..^ |
| 80 | 79 | ralimdva 2962 |
. . . . . . . . . 10
..^ ..^ ..^
..^ ..^ ..^ |
| 81 | 68, 80 | syl5bi 232 |
. . . . . . . . 9
..^ ..^ ..^
..^ ..^ ..^ |
| 82 | 81 | adantrd 484 |
. . . . . . . 8
..^ ..^ ..^
..^ ..^ ..^ ..^ |
| 83 | 82 | adantld 483 |
. . . . . . 7
..^ ..^ ..^ ..^ ..^ ..^ ..^ ..^ |
| 84 | 65, 83 | sylbid 230 |
. . . . . 6
..^ ..^ ..^
..^ ..^ ..^ |
| 85 | 18, 84 | mpd 15 |
. . . . 5
..^ ..^ ..^ ..^ |
| 86 | dff14a 6527 |
. . . . 5
..^..^ ..^..^ ..^ ..^ ..^ ..^ | |
| 87 | 16, 85, 86 | sylanbrc 698 |
. . . 4
..^..^ |
| 88 | df-f1 5893 |
. . . 4
..^..^ ..^..^ ..^ | |
| 89 | 87, 88 | sylib 208 |
. . 3
..^..^
..^ |
| 90 | 89 | simprd 479 |
. 2
..^ |
| 91 | funcnv0 5955 |
. . 3
 | |
| 92 | 19 | nn0zd 11480 |
. . . . . . . . . . . . 13
|
| 93 | peano2zm 11420 |
. . . . . . . . . . . . 13
| |
| 94 | 92, 93 | syl 17 |
. . . . . . . . . . . 12
|
| 95 | 94 | zred 11482 |
. . . . . . . . . . 11
|
| 96 | 1red 10055 |
. . . . . . . . . . 11
| |
| 97 | 95, 96 | lenltd 10183 |
. . . . . . . . . 10
 |
| 98 | 97 | biimpar 502 |
. . . . . . . . 9
|
| 99 | 5, 98 | syl5eqbr 4688 |
. . . . . . . 8
|
| 100 | 1zzd 11408 |
. . . . . . . . . . 11
| |
| 101 | 5, 94 | syl5eqel 2705 |
. . . . . . . . . . 11
|
| 102 | 100, 101 | jca 554 |
. . . . . . . . . 10
|
| 103 | 102 | adantr 481 |
. . . . . . . . 9
|
| 104 | fzon 12489 |
. . . . . . . . . 10
..^ | |
| 105 | 104 | bicomd 213 |
. . . . . . . . 9
..^
|
| 106 | 103, 105 | syl 17 |
. . . . . . . 8
..^
|
| 107 | 99, 106 | mpbird 247 |
. . . . . . 7
..^ |
| 108 | 107 | reseq2d 5396 |
. . . . . 6
..^ |
| 109 | res0 5400 |
. . . . . 6
| |
| 110 | 108, 109 | syl6eq 2672 |
. . . . 5
..^ |
| 111 | 110 | cnveqd 5298 |
. . . 4
..^ |
| 112 | 111 | funeqd 5910 |
. . 3
..^ |
| 113 | 91, 112 | mpbiri 248 |
. 2
..^ |
| 114 | 90, 113 | pm2.61dan 832 |
1
..^ |
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 cvv 3200 cun 3572 wss 3574 c0 3915 csn 4177 class class class wbr 4653
ccnv 5113
cres 5116
wfun 5882
wf 5884 wf1 5885
cfv 5888
(class class class)co 6650 cr 9935 cc0 9936 c1 9937 caddc 9939 clt 10074 cle 10075 cmin 10266 cn 11020 c2 11070 cn0 11292 cz 11377 cuz 11687 ..^cfzo 12465
chash 13117 Word cword 13291 |
| 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 |
| This theorem is referenced by: pthd 26665 |
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