ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xrlttri3 Unicode version

Theorem xrlttri3 8488
Description: Extended real version of lttri3 6895. (Contributed by NM, 9-Feb-2006.)
Assertion
Ref Expression
xrlttri3  RR*  RR*  <  <

Proof of Theorem xrlttri3
StepHypRef Expression
1 elxr 8466 . 2  RR*  RR +oo -oo
2 elxr 8466 . 2  RR*  RR +oo -oo
3 lttri3 6895 . . . . . 6  RR  RR  <  <
43ancoms 255 . . . . 5  RR  RR  <  <
5 renepnf 6870 . . . . . . . . . 10  RR  =/= +oo
65adantr 261 . . . . . . . . 9  RR +oo  =/= +oo
7 neeq2 2214 . . . . . . . . . 10 +oo  =/=  =/= +oo
87adantl 262 . . . . . . . . 9  RR +oo  =/=  =/= +oo
96, 8mpbird 156 . . . . . . . 8  RR +oo  =/=
109necomd 2285 . . . . . . 7  RR +oo  =/=
1110neneqd 2221 . . . . . 6  RR +oo
12 ltpnf 8472 . . . . . . . . 9  RR  < +oo
1312adantr 261 . . . . . . . 8  RR +oo  < +oo
14 breq2 3759 . . . . . . . . 9 +oo  <  < +oo
1514adantl 262 . . . . . . . 8  RR +oo  <  < +oo
1613, 15mpbird 156 . . . . . . 7  RR +oo  <
17 notnot1 559 . . . . . . . . 9  <  <  <  <
1817olcs 654 . . . . . . . 8  <  <  <
19 ioran 668 . . . . . . . 8  <  <  <  <
2018, 19sylnib 600 . . . . . . 7  <  <  <
2116, 20syl 14 . . . . . 6  RR +oo  <  <
2211, 212falsed 617 . . . . 5  RR +oo  <  <
23 renemnf 6871 . . . . . . . . . 10  RR  =/= -oo
2423adantr 261 . . . . . . . . 9  RR -oo  =/= -oo
25 neeq2 2214 . . . . . . . . . 10 -oo  =/=  =/= -oo
2625adantl 262 . . . . . . . . 9  RR -oo  =/=  =/= -oo
2724, 26mpbird 156 . . . . . . . 8  RR -oo  =/=
2827necomd 2285 . . . . . . 7  RR -oo  =/=
2928neneqd 2221 . . . . . 6  RR -oo
30 mnflt 8474 . . . . . . . . 9  RR -oo  <
3130adantr 261 . . . . . . . 8  RR -oo -oo  <
32 breq1 3758 . . . . . . . . 9 -oo  < -oo  <
3332adantl 262 . . . . . . . 8  RR -oo  < -oo  <
3431, 33mpbird 156 . . . . . . 7  RR -oo  <
35 orc 632 . . . . . . 7  <  <  <
36 oranim 806 . . . . . . 7  <  <  <  <
3734, 35, 363syl 17 . . . . . 6  RR -oo  <  <
3829, 372falsed 617 . . . . 5  RR -oo  <  <
394, 22, 383jaodan 1200 . . . 4  RR  RR +oo -oo  <  <
4039ancoms 255 . . 3  RR +oo -oo  RR  <  <
41 renepnf 6870 . . . . . . . . 9  RR  =/= +oo
4241adantl 262 . . . . . . . 8 +oo  RR  =/= +oo
43 neeq2 2214 . . . . . . . . 9 +oo  =/=  =/= +oo
4443adantr 261 . . . . . . . 8 +oo  RR  =/=  =/= +oo
4542, 44mpbird 156 . . . . . . 7 +oo  RR  =/=
4645neneqd 2221 . . . . . 6 +oo  RR
47 ltpnf 8472 . . . . . . . . 9  RR  < +oo
4847adantl 262 . . . . . . . 8 +oo  RR  < +oo
49 breq2 3759 . . . . . . . . 9 +oo  <  < +oo
5049adantr 261 . . . . . . . 8 +oo  RR  <  < +oo
5148, 50mpbird 156 . . . . . . 7 +oo  RR  <
5251, 35, 363syl 17 . . . . . 6 +oo  RR  <  <
5346, 522falsed 617 . . . . 5 +oo  RR  <  <
54 eqtr3 2056 . . . . . . 7 +oo +oo
5554eqcomd 2042 . . . . . 6 +oo +oo
56 pnfxr 8462 . . . . . . . . 9 +oo  RR*
57 xrltnr 8471 . . . . . . . . 9 +oo  RR* +oo  < +oo
5856, 57ax-mp 7 . . . . . . . 8 +oo  < +oo
59 breq12 3760 . . . . . . . . 9 +oo +oo  < +oo  < +oo
6059ancoms 255 . . . . . . . 8 +oo +oo  < +oo  < +oo
6158, 60mtbiri 599 . . . . . . 7 +oo +oo  <
62 breq12 3760 . . . . . . . 8 +oo +oo  < +oo  < +oo
6358, 62mtbiri 599 . . . . . . 7 +oo +oo  <
6461, 63jca 290 . . . . . 6 +oo +oo  <  <
6555, 642thd 164 . . . . 5 +oo +oo  <  <
66 mnfnepnf 8468 . . . . . . . . 9 -oo  =/= +oo
67 eqeq12 2049 . . . . . . . . . 10 -oo +oo -oo +oo
6867necon3bid 2240 . . . . . . . . 9 -oo +oo  =/= -oo  =/= +oo
6966, 68mpbiri 157 . . . . . . . 8 -oo +oo  =/=
7069ancoms 255 . . . . . . 7 +oo -oo  =/=
7170neneqd 2221 . . . . . 6 +oo -oo
72 mnfltpnf 8476 . . . . . . . . 9 -oo  < +oo
73 breq12 3760 . . . . . . . . 9 -oo +oo  < -oo  < +oo
7472, 73mpbiri 157 . . . . . . . 8 -oo +oo  <
7574ancoms 255 . . . . . . 7 +oo -oo  <
7675, 35, 363syl 17 . . . . . 6 +oo -oo  <  <
7771, 762falsed 617 . . . . 5 +oo -oo  <  <
7853, 65, 773jaodan 1200 . . . 4 +oo  RR +oo -oo  <  <
7978ancoms 255 . . 3  RR +oo -oo +oo  <  <
80 renemnf 6871 . . . . . . . . 9  RR  =/= -oo
8180adantl 262 . . . . . . . 8 -oo  RR  =/= -oo
82 neeq2 2214 . . . . . . . . 9 -oo  =/=  =/= -oo
8382adantr 261 . . . . . . . 8 -oo  RR  =/=  =/= -oo
8481, 83mpbird 156 . . . . . . 7 -oo  RR  =/=
8584neneqd 2221 . . . . . 6 -oo  RR
86 mnflt 8474 . . . . . . . . 9  RR -oo  <
8786adantl 262 . . . . . . . 8 -oo  RR -oo  <
88 breq1 3758 . . . . . . . . 9 -oo  < -oo  <
8988adantr 261 . . . . . . . 8 -oo  RR  < -oo  <
9087, 89mpbird 156 . . . . . . 7 -oo  RR  <
9190, 20syl 14 . . . . . 6 -oo  RR  <  <
9285, 912falsed 617 . . . . 5 -oo  RR  <  <
9366neii 2205 . . . . . . . . . 10 -oo +oo
94 eqeq12 2049 . . . . . . . . . 10 -oo +oo -oo +oo
9593, 94mtbiri 599 . . . . . . . . 9 -oo +oo
9695neneqad 2278 . . . . . . . 8 -oo +oo  =/=
9796necomd 2285 . . . . . . 7 -oo +oo  =/=
9897neneqd 2221 . . . . . 6 -oo +oo
99 breq12 3760 . . . . . . . 8 -oo +oo  < -oo  < +oo
10072, 99mpbiri 157 . . . . . . 7 -oo +oo  <
101100, 20syl 14 . . . . . 6 -oo +oo  <  <
10298, 1012falsed 617 . . . . 5 -oo +oo  <  <
103 eqtr3 2056 . . . . . . 7 -oo -oo
104103ancoms 255 . . . . . 6 -oo -oo
105 mnfxr 8464 . . . . . . . . 9 -oo  RR*
106 xrltnr 8471 . . . . . . . . 9 -oo  RR* -oo  < -oo
107105, 106ax-mp 7 . . . . . . . 8 -oo  < -oo
108 breq12 3760 . . . . . . . . 9 -oo -oo  < -oo  < -oo
109108ancoms 255 . . . . . . . 8 -oo -oo  < -oo  < -oo
110107, 109mtbiri 599 . . . . . . 7 -oo -oo  <
111 breq12 3760 . . . . . . . 8 -oo -oo  < -oo  < -oo
112107, 111mtbiri 599 . . . . . . 7 -oo -oo  <
113110, 112jca 290 . . . . . 6 -oo -oo  <  <
114104, 1132thd 164 . . . . 5 -oo -oo  <  <
11592, 102, 1143jaodan 1200 . . . 4 -oo  RR +oo -oo  <  <
116115ancoms 255 . . 3  RR +oo -oo -oo  <  <
11740, 79, 1163jaodan 1200 . 2  RR +oo -oo  RR +oo -oo  <  <
1181, 2, 117syl2anb 275 1  RR*  RR*  <  <
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   w3o 883   wceq 1242   wcel 1390    =/= wne 2201   class class class wbr 3755   RRcr 6710   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856    < clt 6857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795  ax-pre-apti 6798
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862
This theorem is referenced by:  xrletri3  8491  iccid  8564
  Copyright terms: Public domain W3C validator