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Theorem ltxr 6664
Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173.
Assertion
Ref Expression
ltxr |- ((A e. RR* /\ B e. RR*) -> (A < B <-> ((((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo)) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))))
Proof of Theorem ltxr
StepHypRef Expression
1 df-br 3339 . . . . . . . . . 10 |- (A <R B <-> <.A, B>. e. <R )
21bicomi 189 . . . . . . . . 9 |- (<.A, B>. e. <R <-> A <R B)
32a1i 8 . . . . . . . 8 |- (B e. RR* -> (<.A, B>. e. <R <-> A <R B))
4 opelxpg 4039 . . . . . . . 8 |- (B e. RR* -> (<.A, B>. e. (RR X. RR) <-> (A e. RR /\ B e. RR)))
53, 4anbi12d 690 . . . . . . 7 |- (B e. RR* -> ((<.A, B>. e. <R /\ <.A, B>. e. (RR X. RR)) <-> (A <R B /\ (A e. RR /\ B e. RR))))
6 elin 2786 . . . . . . 7 |- (<.A, B>. e. ( <R i^i (RR X. RR)) <-> (<.A, B>. e. <R /\ <.A, B>. e. (RR X. RR)))
7 ancom 482 . . . . . . 7 |- (((A e. RR /\ B e. RR) /\ A <R B) <-> (A <R B /\ (A e. RR /\ B e. RR)))
85, 6, 73bitr4g 614 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. ( <R i^i (RR X. RR)) <-> ((A e. RR /\ B e. RR) /\ A <R B)))
98adantl 424 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. ( <R i^i (RR X. RR)) <-> ((A e. RR /\ B e. RR) /\ A <R B)))
10 pnfxr 6660 . . . . . . 7 |- +oo e. RR*
11 opthgg 3534 . . . . . . 7 |- ((A e. RR* /\ B e. RR* /\ +oo e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
1210, 11mp3an3 1180 . . . . . 6 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
13 opex 3527 . . . . . . 7 |- <.A, B>. e. _V
1413elsnc 3065 . . . . . 6 |- (<.A, B>. e. {<. -oo, +oo>.} <-> <.A, B>. = <. -oo, +oo>.)
1512, 14syl5bb 591 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. {<. -oo, +oo>.} <-> (A = -oo /\ B = +oo)))
169, 15orbi12d 689 . . . 4 |- ((A e. RR* /\ B e. RR*) -> ((<.A, B>. e. ( <R i^i (RR X. RR)) \/ <.A, B>. e. {<. -oo, +oo>.}) <-> (((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo))))
17 elun 2741 . . . 4 |- (<.A, B>. e. (( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) <-> (<.A, B>. e. ( <R i^i (RR X. RR)) \/ <.A, B>. e. {<. -oo, +oo>.}))
1816, 17syl5bb 591 . . 3 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. (( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) <-> (((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo))))
19 opelxpg 4039 . . . . . . 7 |- (B e. RR* -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B e. { +oo})))
20 elsncg 3063 . . . . . . . 8 |- (B e. RR* -> (B e. { +oo} <-> B = +oo))
2120anbi2d 678 . . . . . . 7 |- (B e. RR* -> ((A e. RR /\ B e. { +oo}) <-> (A e. RR /\ B = +oo)))
2219, 21bitrd 587 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B = +oo)))
2322adantl 424 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B = +oo)))
24 opelxpg 4039 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. ({ -oo} X. RR) <-> (A e. { -oo} /\ B e. RR)))
25 elsncg 3063 . . . . . . 7 |- (A e. RR* -> (A e. { -oo} <-> A = -oo))
2625anbi1d 679 . . . . . 6 |- (A e. RR* -> ((A e. { -oo} /\ B e. RR) <-> (A = -oo /\ B e. RR)))
2724, 26sylan9bbr 600 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. ({ -oo} X. RR) <-> (A = -oo /\ B e. RR)))
2823, 27orbi12d 689 . . . 4 |- ((A e. RR* /\ B e. RR*) -> ((<.A, B>. e. (RR X. { +oo}) \/ <.A, B>. e. ({ -oo} X. RR)) <-> ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))))
29 elun 2741 . . . 4 |- (<.A, B>. e. ((RR X. { +oo}) u. ({ -oo} X. RR)) <-> (<.A, B>. e. (RR X. { +oo}) \/ <.A, B>. e. ({ -oo} X. RR)))
3028, 29syl5bb 591 . . 3 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. ((RR X. { +oo}) u. ({ -oo} X. RR)) <-> ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))))
3118, 30orbi12d 689 . 2 |- ((A e. RR* /\ B e. RR*) -> ((<.A, B>. e. (( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) \/ <.A, B>. e. ((RR X. { +oo}) u. ({ -oo} X. RR))) <-> ((((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo)) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))))
32 df-br 3339 . . 3 |- (A < B <-> <.A, B>. e. < )
33 df-ltxr 6657 . . . 4 |- < = ((( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) u. ((RR X. { +oo}) u. ({ -oo} X. RR)))
3433eleq2i 1961 . . 3 |- (<.A, B>. e. < <-> <.A, B>. e. ((( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) u. ((RR X. { +oo}) u. ({ -oo} X. RR))))
35 elun 2741 . . 3 |- (<.A, B>. e. ((( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) u. ((RR X. { +oo}) u. ({ -oo} X. RR))) <-> (<.A, B>. e. (( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) \/ <.A, B>. e. ((RR X. { +oo}) u. ({ -oo} X. RR))))
3632, 34, 353bitri 194 . 2 |- (A < B <-> (<.A, B>. e. (( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) \/ <.A, B>. e. ((RR X. { +oo}) u. ({ -oo} X. RR))))
3731, 36syl5bb 591 1 |- ((A e. RR* /\ B e. RR*) -> (A < B <-> ((((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo)) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   u. cun 2591   i^i cin 2592  {csn 3044  <.cop 3046   class class class wbr 3338   X. cxp 3984  RRcr 6385   <R cltrr 6390   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652   < clt 6653
This theorem is referenced by:  ltxrlt 6669  xrltnr 6716  ltpnf 6717  mnflt 6718  mnfltpnf 6719  pnfnlt 6721  nltmnf 6722
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-qs 5323  df-ni 6152  df-nq 6190  df-np 6238  df-nr 6319  df-c 6392  df-pnf 6654  df-xr 6656  df-ltxr 6657
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