Nearby theorems
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Mathbox for Alexander van der Vekens |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > icceuelpart | Structured version Visualization version Unicode version | ||
| Description: An element of a partitioned half opened interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.) |
| Ref | Expression |
|---|---|
| iccpartiun.m |
        |
| iccpartiun.p |
 RePart |
| Ref | Expression |
|---|---|
| icceuelpart |
![/\](wedge.gif "/\"){kind=small}
    
..^   |
, , , ,
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iccpartiun.p |
. . . 4
RePart | |
| 2 | 1 | adantr 481 |
. . 3
RePart |
| 3 | iccpartiun.m |
. . . . 5
| |
| 4 | iccelpart 41369 |
. . . . 5
 RePart  ..^ | |
| 5 | 3, 4 | syl 17 |
. . . 4
RePart ..^ |
| 6 | 5 | adantr 481 |
. . 3
RePart ..^ |
| 7 | fveq1 6190 |
. . . . . . . . 9
 | |
| 8 | fveq1 6190 |
. . . . . . . . 9
| |
| 9 | 7, 8 | oveq12d 6668 |
. . . . . . . 8
|
| 10 | 9 | eleq2d 2687 |
. . . . . . 7
|
| 11 | fveq1 6190 |
. . . . . . . . . 10
| |
| 12 | fveq1 6190 |
. . . . . . . . . 10
| |
| 13 | 11, 12 | oveq12d 6668 |
. . . . . . . . 9
|
| 14 | 13 | eleq2d 2687 |
. . . . . . . 8
|
| 15 | 14 | rexbidv 3052 |
. . . . . . 7
..^
..^ |
| 16 | 10, 15 | imbi12d 334 |
. . . . . 6
..^
..^ |
| 17 | 16 | rspcva 3307 |
. . . . 5
RePart
RePart
..^
..^ |
| 18 | 17 | adantld 483 |
. . . 4
RePart
RePart
..^
..^ |
| 19 | 18 | com12 32 |
. . 3
RePart
RePart
..^
..^ |
| 20 | 2, 6, 19 | mp2and 715 |
. 2
..^ |
| 21 | 3 | adantr 481 |
. . . . . . . . . 10
..^ |
| 22 | 1 | adantr 481 |
. . . . . . . . . 10
..^ RePart |
| 23 | elfzofz 12485 |
. . . . . . . . . . 11
..^
 | |
| 24 | 23 | adantl 482 |
. . . . . . . . . 10
..^ |
| 25 | 21, 22, 24 | iccpartxr 41355 |
. . . . . . . . 9
..^  |
| 26 | fzofzp1 12565 |
. . . . . . . . . . 11
..^
| |
| 27 | 26 | adantl 482 |
. . . . . . . . . 10
..^ |
| 28 | 21, 22, 27 | iccpartxr 41355 |
. . . . . . . . 9
..^ |
| 29 | 25, 28 | jca 554 |
. . . . . . . 8
..^
|
| 30 | 29 | adantrr 753 |
. . . . . . 7
..^ ..^
|
| 31 | elico1 12218 |
. . . . . . 7
  | |
| 32 | 30, 31 | syl 17 |
. . . . . 6
..^ ..^
|
| 33 | 3 | adantr 481 |
. . . . . . . . . 10
..^ |
| 34 | 1 | adantr 481 |
. . . . . . . . . 10
..^ RePart |
| 35 | elfzofz 12485 |
. . . . . . . . . . 11
..^
| |
| 36 | 35 | adantl 482 |
. . . . . . . . . 10
..^ |
| 37 | 33, 34, 36 | iccpartxr 41355 |
. . . . . . . . 9
..^ |
| 38 | fzofzp1 12565 |
. . . . . . . . . . 11
..^
| |
| 39 | 38 | adantl 482 |
. . . . . . . . . 10
..^ |
| 40 | 33, 34, 39 | iccpartxr 41355 |
. . . . . . . . 9
..^ |
| 41 | 37, 40 | jca 554 |
. . . . . . . 8
..^
|
| 42 | 41 | adantrl 752 |
. . . . . . 7
..^ ..^
|
| 43 | elico1 12218 |
. . . . . . 7
| |
| 44 | 42, 43 | syl 17 |
. . . . . 6
..^ ..^
|
| 45 | 32, 44 | anbi12d 747 |
. . . . 5
..^ ..^
|
| 46 | elfzoelz 12470 |
. . . . . . . . . 10
..^
 | |
| 47 | 46 | zred 11482 |
. . . . . . . . 9
..^
 |
| 48 | elfzoelz 12470 |
. . . . . . . . . 10
..^
| |
| 49 | 48 | zred 11482 |
. . . . . . . . 9
..^
|
| 50 | 47, 49 | anim12i 590 |
. . . . . . . 8
..^ ..^
|
| 51 | 50 | adantl 482 |
. . . . . . 7
..^ ..^
|
| 52 | lttri4 10122 |
. . . . . . 7
| |
| 53 | 51, 52 | syl 17 |
. . . . . 6
..^ ..^
|
| 54 | 3, 1 | icceuelpartlem 41371 |
. . . . . . . . . 10
..^ ..^
|
| 55 | 54 | imp31 448 |
. . . . . . . . 9
..^ ..^ |
| 56 | simpl 473 |
. . . . . . . . . . . . . . . . . . . . . 22
..^ ..^ | |
| 57 | 28 | adantrr 753 |
. . . . . . . . . . . . . . . . . . . . . . 23
..^ ..^
|
| 58 | 57 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
..^ ..^ |
| 59 | 37 | adantrl 752 |
. . . . . . . . . . . . . . . . . . . . . . 23
..^ ..^
|
| 60 | 59 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
..^ ..^ |
| 61 | nltle2tri 41323 |
. . . . . . . . . . . . . . . . . . . . . 22
| |
| 62 | 56, 58, 60, 61 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . . 21
..^ ..^
|
| 63 | 62 | pm2.21d 118 |
. . . . . . . . . . . . . . . . . . . 20
..^ ..^
|
| 64 | 63 | 3expd 1284 |
. . . . . . . . . . . . . . . . . . 19
..^ ..^
|
| 65 | 64 | ex 450 |
. . . . . . . . . . . . . . . . . 18
..^ ..^
|
| 66 | 65 | com23 86 |
. . . . . . . . . . . . . . . . 17
..^ ..^
|
| 67 | 66 | com25 99 |
. . . . . . . . . . . . . . . 16
..^ ..^
|
| 68 | 67 | imp4b 613 |
. . . . . . . . . . . . . . 15
..^
..^
|
| 69 | 68 | com23 86 |
. . . . . . . . . . . . . 14
..^
..^
|
| 70 | 69 | 3adant3 1081 |
. . . . . . . . . . . . 13
..^
..^
|
| 71 | 70 | com12 32 |
. . . . . . . . . . . 12
..^
..^
|
| 72 | 71 | 3ad2ant3 1084 |
. . . . . . . . . . 11
..^
..^
|
| 73 | 72 | imp 445 |
. . . . . . . . . 10
..^
..^
|
| 74 | 73 | com12 32 |
. . . . . . . . 9
..^ ..^
|
| 75 | 55, 74 | syldan 487 |
. . . . . . . 8
..^ ..^
|
| 76 | 75 | expcom 451 |
. . . . . . 7
..^ ..^
|
| 77 | 2a1 28 |
. . . . . . 7
..^ ..^
| |
| 78 | 3, 1 | icceuelpartlem 41371 |
. . . . . . . . . . 11
..^ ..^
|
| 79 | 78 | ancomsd 470 |
. . . . . . . . . 10
..^ ..^
|
| 80 | 79 | imp31 448 |
. . . . . . . . 9
..^ ..^ |
| 81 | 40 | adantrl 752 |
. . . . . . . . . . . . . . . . . . . . . 22
..^ ..^
|
| 82 | 81 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
..^ ..^ |
| 83 | 25 | adantrr 753 |
. . . . . . . . . . . . . . . . . . . . . 22
..^ ..^
|
| 84 | 83 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
..^ ..^ |
| 85 | nltle2tri 41323 |
. . . . . . . . . . . . . . . . . . . . 21
| |
| 86 | 56, 82, 84, 85 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . 20
..^ ..^
|
| 87 | 86 | pm2.21d 118 |
. . . . . . . . . . . . . . . . . . 19
..^ ..^
|
| 88 | 87 | 3expd 1284 |
. . . . . . . . . . . . . . . . . 18
..^ ..^
|
| 89 | 88 | ex 450 |
. . . . . . . . . . . . . . . . 17
..^ ..^
|
| 90 | 89 | com23 86 |
. . . . . . . . . . . . . . . 16
..^ ..^
|
| 91 | 90 | imp4b 613 |
. . . . . . . . . . . . . . 15
..^
..^
|
| 92 | 91 | com23 86 |
. . . . . . . . . . . . . 14
..^
..^
|
| 93 | 92 | 3adant2 1080 |
. . . . . . . . . . . . 13
..^
..^
|
| 94 | 93 | com12 32 |
. . . . . . . . . . . 12
..^
..^
|
| 95 | 94 | 3ad2ant2 1083 |
. . . . . . . . . . 11
..^
..^
|
| 96 | 95 | imp 445 |
. . . . . . . . . 10
..^
..^
|
| 97 | 96 | com12 32 |
. . . . . . . . 9
..^ ..^
|
| 98 | 80, 97 | syldan 487 |
. . . . . . . 8
..^ ..^
|
| 99 | 98 | expcom 451 |
. . . . . . 7
..^ ..^
|
| 100 | 76, 77, 99 | 3jaoi 1391 |
. . . . . 6
..^
..^
|
| 101 | 53, 100 | mpcom 38 |
. . . . 5
..^ ..^
|
| 102 | 45, 101 | sylbid 230 |
. . . 4
..^ ..^
|
| 103 | 102 | ralrimivva 2971 |
. . 3
..^
..^
|
| 104 | 103 | adantr 481 |
. 2
..^ ..^
|
| 105 | fveq2 6191 |
. . . . 5
| |
| 106 | oveq1 6657 |
. . . . . 6
| |
| 107 | 106 | fveq2d 6195 |
. . . . 5
|
| 108 | 105, 107 | oveq12d 6668 |
. . . 4
|
| 109 | 108 | eleq2d 2687 |
. . 3
|
| 110 | 109 | reu4 3400 |
. 2
..^ ..^ ..^ ..^ |
| 111 | 20, 104, 110 | sylanbrc 698 |
1
..^ |
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wb 196 wa 384 w3o 1036 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wreu 2914 class class class wbr 4653
cfv 5888
(class class class)co 6650 cr 9935 cc0 9936 c1 9937 caddc 9939 cxr 10073 clt 10074 cle 10075 cn 11020 cico 12177 cfz 12326 ..^cfzo 12465
RePartciccp 41349 |
| 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-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-fal 1489 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-rmo 2920 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-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-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 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-ico 12181 df-fz 12327 df-fzo 12466 df-iccp 41350 |
| This theorem is referenced by: iccpartdisj 41373 |
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