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Theorem ltxrt 5478
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 2616 . . . . . . . . . 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 3212 . . . . . . . 8 |- (B e. RR* -> (<.A, B>. e. (RR X. RR) <-> (A e. RR /\ B e. RR)))
53, 4anbi12d 627 . . . . . . 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 2204 . . . . . . 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 554 . . . . . 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 5476 . . . . . . 7 |- +oo e. RR*
11 opthgg 2785 . . . . . . 7 |- ((A e. RR* /\ B e. RR* /\ +oo e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
1210, 11mp3an3 904 . . . . . 6 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
13 opex 2778 . . . . . . 7 |- <.A, B>. e. V
1413elsnc 2428 . . . . . 6 |- (<.A, B>. e. {<. -oo, +oo>.} <-> <.A, B>. = <. -oo, +oo>.)
1512, 14syl5bb 531 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. {<. -oo, +oo>.} <-> (A = -oo /\ B = +oo)))
169, 15orbi12d 626 . . . 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 2170 . . . 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 531 . . 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 3212 . . . . . . 7 |- (B e. RR* -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B e. { +oo})))
20 elsncg 2427 . . . . . . . 8 |- (B e. RR* -> (B e. { +oo} <-> B = +oo))
2120anbi2d 615 . . . . . . 7 |- (B e. RR* -> ((A e. RR /\ B e. { +oo}) <-> (A e. RR /\ B = +oo)))
2219, 21bitrd 527 . . . . . 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 3212 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. ({ -oo} X. RR) <-> (A e. { -oo} /\ B e. RR)))
25 elsncg 2427 . . . . . . 7 |- (A e. RR* -> (A e. { -oo} <-> A = -oo))
2625anbi1d 616 . . . . . 6 |- (A e. RR* -> ((A e. { -oo} /\ B e. RR) <-> (A = -oo /\ B e. RR)))
2724, 26sylan9bbr 540 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. ({ -oo} X. RR) <-> (A = -oo /\ B e. RR)))
2823, 27orbi12d 626 . . . 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 2170 . . . 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 531 . . 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 626 . 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 2616 . . 3 |- (A < B <-> <.A, B>. e. < )
33 df-ltxr 5473 . . . 4 |- < = ((( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) u. ((RR X. { +oo}) u. ({ -oo} X. RR)))
3433eleq2i 1536 . . 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 2170 . . 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 531 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 955   e. wcel 957   u. cun 2042   i^i cin 2043  {csn 2406  <.cop 2408   class class class wbr 2615   X. cxp 3164  RRcr 5216   <R cltrr 5221   +oocpnf 5466   -oocmnf 5467  RR*cxr 5468   < clt 5469
This theorem is referenced by:  ltxrltt 5483  xrltnrt 5524  ltpnft 5525  mnfltt 5526  mnfltpnf 5527  pnfnltt 5529  nltmnft 5530
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-qs 4259  df-ni 4983  df-nq 5021  df-np 5069  df-nr 5150  df-c 5223  df-pnf 5470  df-xr 5472  df-ltxr 5473
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