Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > qbtwnxr | Structured version Visualization version Unicode version |
Description: The rational numbers are dense in : any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
qbtwnxr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 11950 | . . 3 | |
2 | elxr 11950 | . . . . 5 | |
3 | qbtwnre 12030 | . . . . . . 7 | |
4 | 3 | 3expia 1267 | . . . . . 6 |
5 | simpl 473 | . . . . . . . . 9 | |
6 | peano2re 10209 | . . . . . . . . . 10 | |
7 | 6 | adantr 481 | . . . . . . . . 9 |
8 | ltp1 10861 | . . . . . . . . . 10 | |
9 | 8 | adantr 481 | . . . . . . . . 9 |
10 | qbtwnre 12030 | . . . . . . . . 9 | |
11 | 5, 7, 9, 10 | syl3anc 1326 | . . . . . . . 8 |
12 | qre 11793 | . . . . . . . . . . . . . 14 | |
13 | ltpnf 11954 | . . . . . . . . . . . . . 14 | |
14 | 12, 13 | syl 17 | . . . . . . . . . . . . 13 |
15 | 14 | adantl 482 | . . . . . . . . . . . 12 |
16 | simplr 792 | . . . . . . . . . . . 12 | |
17 | 15, 16 | breqtrrd 4681 | . . . . . . . . . . 11 |
18 | 17 | a1d 25 | . . . . . . . . . 10 |
19 | 18 | anim2d 589 | . . . . . . . . 9 |
20 | 19 | reximdva 3017 | . . . . . . . 8 |
21 | 11, 20 | mpd 15 | . . . . . . 7 |
22 | 21 | a1d 25 | . . . . . 6 |
23 | rexr 10085 | . . . . . . 7 | |
24 | breq2 4657 | . . . . . . . . 9 | |
25 | 24 | adantl 482 | . . . . . . . 8 |
26 | nltmnf 11963 | . . . . . . . . . 10 | |
27 | 26 | adantr 481 | . . . . . . . . 9 |
28 | 27 | pm2.21d 118 | . . . . . . . 8 |
29 | 25, 28 | sylbid 230 | . . . . . . 7 |
30 | 23, 29 | sylan 488 | . . . . . 6 |
31 | 4, 22, 30 | 3jaodan 1394 | . . . . 5 |
32 | 2, 31 | sylan2b 492 | . . . 4 |
33 | breq1 4656 | . . . . . 6 | |
34 | 33 | adantr 481 | . . . . 5 |
35 | pnfnlt 11962 | . . . . . . 7 | |
36 | 35 | adantl 482 | . . . . . 6 |
37 | 36 | pm2.21d 118 | . . . . 5 |
38 | 34, 37 | sylbid 230 | . . . 4 |
39 | peano2rem 10348 | . . . . . . . . . 10 | |
40 | 39 | adantl 482 | . . . . . . . . 9 |
41 | simpr 477 | . . . . . . . . 9 | |
42 | ltm1 10863 | . . . . . . . . . 10 | |
43 | 42 | adantl 482 | . . . . . . . . 9 |
44 | qbtwnre 12030 | . . . . . . . . 9 | |
45 | 40, 41, 43, 44 | syl3anc 1326 | . . . . . . . 8 |
46 | simpll 790 | . . . . . . . . . . . 12 | |
47 | 12 | adantl 482 | . . . . . . . . . . . . 13 |
48 | mnflt 11957 | . . . . . . . . . . . . 13 | |
49 | 47, 48 | syl 17 | . . . . . . . . . . . 12 |
50 | 46, 49 | eqbrtrd 4675 | . . . . . . . . . . 11 |
51 | 50 | a1d 25 | . . . . . . . . . 10 |
52 | 51 | anim1d 588 | . . . . . . . . 9 |
53 | 52 | reximdva 3017 | . . . . . . . 8 |
54 | 45, 53 | mpd 15 | . . . . . . 7 |
55 | 54 | a1d 25 | . . . . . 6 |
56 | 1re 10039 | . . . . . . . . . 10 | |
57 | mnflt 11957 | . . . . . . . . . 10 | |
58 | 56, 57 | ax-mp 5 | . . . . . . . . 9 |
59 | breq1 4656 | . . . . . . . . 9 | |
60 | 58, 59 | mpbiri 248 | . . . . . . . 8 |
61 | ltpnf 11954 | . . . . . . . . . 10 | |
62 | 56, 61 | ax-mp 5 | . . . . . . . . 9 |
63 | breq2 4657 | . . . . . . . . 9 | |
64 | 62, 63 | mpbiri 248 | . . . . . . . 8 |
65 | 1z 11407 | . . . . . . . . . 10 | |
66 | zq 11794 | . . . . . . . . . 10 | |
67 | 65, 66 | ax-mp 5 | . . . . . . . . 9 |
68 | breq2 4657 | . . . . . . . . . . 11 | |
69 | breq1 4656 | . . . . . . . . . . 11 | |
70 | 68, 69 | anbi12d 747 | . . . . . . . . . 10 |
71 | 70 | rspcev 3309 | . . . . . . . . 9 |
72 | 67, 71 | mpan 706 | . . . . . . . 8 |
73 | 60, 64, 72 | syl2an 494 | . . . . . . 7 |
74 | 73 | a1d 25 | . . . . . 6 |
75 | 3mix3 1232 | . . . . . . . 8 | |
76 | 75, 1 | sylibr 224 | . . . . . . 7 |
77 | 76, 29 | sylan 488 | . . . . . 6 |
78 | 55, 74, 77 | 3jaodan 1394 | . . . . 5 |
79 | 2, 78 | sylan2b 492 | . . . 4 |
80 | 32, 38, 79 | 3jaoian 1393 | . . 3 |
81 | 1, 80 | sylanb 489 | . 2 |
82 | 81 | 3impia 1261 | 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 wrex 2913 class class class wbr 4653 (class class class)co 6650 cr 9935 c1 9937 caddc 9939 cpnf 10071 cmnf 10072 cxr 10073 clt 10074 cmin 10266 cz 11377 cq 11788 |
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 ax-pre-sup 10014 |
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-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-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 |
This theorem is referenced by: qextltlem 12033 xralrple 12036 ixxub 12196 ixxlb 12197 ioo0 12200 ico0 12221 ioc0 12222 blssps 22229 blss 22230 blcld 22310 qdensere 22573 tgqioo 22603 dvlip2 23758 lhop2 23778 itgsubst 23812 itg2gt0cn 33465 qinioo 39762 qelioo 39773 qndenserrnbllem 40514 |
Copyright terms: Public domain | W3C validator |