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Theorem ltxrt 5495
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 2620 . . . . . . . . . 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 3216 . . . . . . . 8 |- (B e. RR* -> (<.A, B>. e. (RR X. RR) <-> (A e. RR /\ B e. RR)))
53, 4anbi12d 628 . . . . . . 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 2207 . . . . . . 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 555 . . . . . 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 5493 . . . . . . 7 |- +oo e. RR*
11 opthgg 2789 . . . . . . 7 |- ((A e. RR* /\ B e. RR* /\ +oo e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
1210, 11mp3an3 905 . . . . . 6 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. = <. -oo, +oo>. <-> (A = -oo /\ B = +oo)))
13 opex 2782 . . . . . . 7 |- <.A, B>. e. V
1413elsnc 2431 . . . . . 6 |- (<.A, B>. e. {<. -oo, +oo>.} <-> <.A, B>. = <. -oo, +oo>.)
1512, 14syl5bb 532 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. {<. -oo, +oo>.} <-> (A = -oo /\ B = +oo)))
169, 15orbi12d 627 . . . 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 2173 . . . 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 532 . . 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 3216 . . . . . . 7 |- (B e. RR* -> (<.A, B>. e. (RR X. { +oo}) <-> (A e. RR /\ B e. { +oo})))
20 elsncg 2430 . . . . . . . 8 |- (B e. RR* -> (B e. { +oo} <-> B = +oo))
2120anbi2d 616 . . . . . . 7 |- (B e. RR* -> ((A e. RR /\ B e. { +oo}) <-> (A e. RR /\ B = +oo)))
2219, 21bitrd 528 . . . . . 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 3216 . . . . . 6 |- (B e. RR* -> (<.A, B>. e. ({ -oo} X. RR) <-> (A e. { -oo} /\ B e. RR)))
25 elsncg 2430 . . . . . . 7 |- (A e. RR* -> (A e. { -oo} <-> A = -oo))
2625anbi1d 617 . . . . . 6 |- (A e. RR* -> ((A e. { -oo} /\ B e. RR) <-> (A = -oo /\ B e. RR)))
2724, 26sylan9bbr 541 . . . . 5 |- ((A e. RR* /\ B e. RR*) -> (<.A, B>. e. ({ -oo} X. RR) <-> (A = -oo /\ B e. RR)))
2823, 27orbi12d 627 . . . 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 2173 . . . 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 532 . . 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 627 . 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 2620 . . 3 |- (A < B <-> <.A, B>. e. < )
33 df-ltxr 5490 . . . 4 |- < = ((( <R i^i (RR X. RR)) u. {<. -oo, +oo>.}) u. ((RR X. { +oo}) u. ({ -oo} X. RR)))
3433eleq2i 1538 . . 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 2173 . . 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 532 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 956   e. wcel 958   u. cun 2045   i^i cin 2046  {csn 2409  <.cop 2411   class class class wbr 2619   X. cxp 3168  RRcr 5233   <R cltrr 5238   +oocpnf 5483   -oocmnf 5484  RR*cxr 5485   < clt 5486
This theorem is referenced by:  ltxrltt 5500  xrltnrt 5541  ltpnft 5542  mnfltt 5543  mnfltpnf 5544  pnfnltt 5546  nltmnft 5547
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-qs 4266  df-ni 5000  df-nq 5038  df-np 5086  df-nr 5167  df-c 5240  df-pnf 5487  df-xr 5489  df-ltxr 5490
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