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Theorem ltxrt 5467
Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173.
Assertion
Ref Expression
ltxrt |- ((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 ltxrt
StepHypRef Expression
1 df-br 2610 . . . . . . . . . 10 |- (A <R B <-> <.A, B>. e. <R )
21bicomi 172 . . . . . . . . 9 |- (<.A, B>. e. <R <-> A <R B)
32a1i 8 . . . . . . . 8 |- (B e. RR* -> (<.A, B>. e. <R <-> A <R B))
4 opelxpg 3206 . . . . . . . 8 |- (B e. RR* -> (<.A, B>. e. (RR X. RR) <-> (A e. RR /\ B e. RR)))
53, 4anbi12d 626 . . . . . . 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 2197 . . . . . . 7 |- (<.A, B>. e. ( <R i^i (RR X. RR)) <-> (<.A, B>. e. <R /\ <.A, B>. e. (RR X. RR)))
7 ancom 435 . . . . . . 7 |- (((A e. RR /\ B e. RR) /\ A <R B) <-> (A <R B /\ (A e. RR /\ B e. RR)))
85, 6, 73bitr4g 553 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. ( <R i^i (RR X. RR)) <-> ((A e. RR /\ B e. RR) /\ A <R B)))
98adantl 388 . . . . 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 5465 . . . . . . 7 |- +oo e. RR*
11 opthgg 2779 . . . . . . 7 |- ((A e. RR* /\ B e. RR* /\ +oo e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
1210, 11mp3an3 902 . . . . . 6 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
13 opex 2772 . . . . . . 7 |- <.A, B>. e. V
1413elsnc 2421 . . . . . 6 |- (<.A, B>. e. {<. -oo, +oo>.} <-> <.A, B>. = <. -oo, +oo>.)
1512, 14syl5bb 530 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. {<. -oo, +oo>.} <-> (A = -oo /\ B = +oo)))
169, 15orbi12d 625 . . . 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 2163 . . . 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 530 . . 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 3206 . . . . . . 7 |- (B e. RR* -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B e. { +oo})))
20 elsncg 2420 . . . . . . . 8 |- (B e. RR* -> (B e. { +oo} <-> B = +oo))
2120anbi2d 614 . . . . . . 7 |- (B e. RR* -> ((A e. RR /\ B e. { +oo}) <-> (A e. RR /\ B = +oo)))
2219, 21bitrd 526 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B = +oo)))
2322adantl 388 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B = +oo)))
24 opelxpg 3206 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. ({ -oo} X. RR) <-> (A e. { -oo} /\ B e. RR)))
25 elsncg 2420 . . . . . . 7 |- (A e. RR* -> (A e. { -oo} <-> A = -oo))
2625anbi1d 615 . . . . . 6 |- (A e. RR* -> ((A e. { -oo} /\ B e. RR) <-> (A = -oo /\ B e. RR)))
2724, 26sylan9bbr 539 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. ({ -oo} X. RR) <-> (A = -oo /\ B e. RR)))
2823, 27orbi12d 625 . . . 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 2163 . . . 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 530 . . 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 625 . 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 2610 . . 3 |- (A < B <-> <.A, B>. e. < )
33 df-ltxr 5462 . . . 4 |- < = ((( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) u. ((RR X. { +oo}) u. ({ -oo} X. RR)))
3433eleq2i 1530 . . 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 2163 . . 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, 353bitr 177 . 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 530 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 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   u. cun 2035   i^i cin 2036  {csn 2399  <.cop 2401   class class class wbr 2609   X. cxp 3158  RRcr 5205   <R cltrr 5210   +oocpnf 5455   -oocmnf 5456  RR*cxr 5457   < clt 5458
This theorem is referenced by:  ltxrltt 5472  xrltnrt 5514  ltpnft 5515  mnfltt 5516  mnfltpnf 5517  pnfnltt 5519  nltmnft 5520
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-qs 4250  df-ni 4972  df-nq 5010  df-np 5058  df-nr 5139  df-c 5212  df-pnf 5459  df-xr 5461  df-ltxr 5462
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