Theorem ltxrltt 5483

Description: The standard less-than <R and the extended real less-than < are identical when restricted to the non-extended reals RR.

Assertion

Ref	Expression
ltxrltt	|- ((A e. RR /\ B e. RR) -> (A < B <-> A <R B))

Proof of Theorem ltxrltt

Step	Hyp	Ref	Expression
1		ltxrt 5478	. . 3 |- ((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)))))
2		rexrt 5482	. . 3 |- (A e. RR -> A e. RR*)
3		rexrt 5482	. . 3 |- (B e. RR -> B e. RR*)
4	1, 2, 3	syl2an 454	. 2 |- ((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)))))
5		ibar 642	. . . 4 |- ((A e. RR /\ B e. RR) -> (A <R B <-> ((A e. RR /\ B e. RR) /\ A <R B)))
6		pnfnre 5479	. . . . . . . . . . . 12 |- +oo e/ RR
7		df-nel 1586	. . . . . . . . . . . 12 |- ( +oo e/ RR <-> -. +oo e. RR)
8	6, 7	mpbi 189	. . . . . . . . . . 11 |- -. +oo e. RR
9		eleq1 1532	. . . . . . . . . . 11 |- (B = +oo -> (B e. RR <-> +oo e. RR))
10	8, 9	mtbiri 716	. . . . . . . . . 10 |- (B = +oo -> -. B e. RR)
11	10	con2i 97	. . . . . . . . 9 |- (B e. RR -> -. B = +oo)
12	11	intnand 692	. . . . . . . 8 |- (B e. RR -> -. (A = -oo /\ B = +oo))
13	12	adantl 388	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> -. (A = -oo /\ B = +oo))
14	11	intnand 692	. . . . . . . . . 10 |- (B e. RR -> -. (A e. RR /\ B = +oo))
15		mnfnre 5480	. . . . . . . . . . . . . 14 |- -oo e/ RR
16		df-nel 1586	. . . . . . . . . . . . . 14 |- ( -oo e/ RR <-> -. -oo e. RR)
17	15, 16	mpbi 189	. . . . . . . . . . . . 13 |- -. -oo e. RR
18		eleq1 1532	. . . . . . . . . . . . 13 |- (A = -oo -> (A e. RR <-> -oo e. RR))
19	17, 18	mtbiri 716	. . . . . . . . . . . 12 |- (A = -oo -> -. A e. RR)
20	19	con2i 97	. . . . . . . . . . 11 |- (A e. RR -> -. A = -oo)
21	20	intnanrd 693	. . . . . . . . . 10 |- (A e. RR -> -. (A = -oo /\ B e. RR))
22	14, 21	anim12i 333	. . . . . . . . 9 |- ((B e. RR /\ A e. RR) -> (-. (A e. RR /\ B = +oo) /\ -. (A = -oo /\ B e. RR)))
23		ioran 306	. . . . . . . . 9 |- (-. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)) <-> (-. (A e. RR /\ B = +oo) /\ -. (A = -oo /\ B e. RR)))
24	22, 23	sylibr 200	. . . . . . . 8 |- ((B e. RR /\ A e. RR) -> -. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))
25	24	ancoms 436	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> -. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))
26	13, 25	jca 288	. . . . . 6 |- ((A e. RR /\ B e. RR) -> (-. (A = -oo /\ B = +oo) /\ -. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))))
27		ioran 306	. . . . . 6 |- (-. ((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) <-> (-. (A = -oo /\ B = +oo) /\ -. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))))
28	26, 27	sylibr 200	. . . . 5 |- ((A e. RR /\ B e. RR) -> -. ((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))))
29		biorf 734	. . . . 5 |- (-. ((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) -> (((A e. RR /\ B e. RR) /\ A <R B) <-> (((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) \/ ((A e. RR /\ B e. RR) /\ A <R B))))
30	28, 29	syl 10	. . . 4 |- ((A e. RR /\ B e. RR) -> (((A e. RR /\ B e. RR) /\ A <R B) <-> (((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) \/ ((A e. RR /\ B e. RR) /\ A <R B))))
31	5, 30	bitr2d 528	. . 3 |- ((A e. RR /\ B e. RR) -> ((((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) \/ ((A e. RR /\ B e. RR) /\ A <R B)) <-> A <R B))
32		orass 260	. . . 4 |- (((((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo)) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) <-> (((A e. RR /\ B e. RR) /\ A <R B) \/ ((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))))
33		orcom 246	. . . 4 |- ((((A e. RR /\ B e. RR) /\ A <R B) \/ ((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))) <-> (((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) \/ ((A e. RR /\ B e. RR) /\ A <R B)))
34	32, 33	bitr 173	. . 3 |- (((((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo)) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) <-> (((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) \/ ((A e. RR /\ B e. RR) /\ A <R B)))
35	31, 34	syl5bb 531	. 2 |- ((A e. RR /\ B e. RR) -> (((((A e. RR /\ B e. RR) /\ A <R B) \/ (A = -oo /\ B = +oo)) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))) <-> A <R B))
36	4, 35	bitrd 527	1 |- ((A e. RR /\ B e. RR) -> (A < B <-> A <R B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   e. wcel 957   e/ wnel 1584   class class class wbr 2615  RRcr 5216   <R cltrr 5221   +oocpnf 5466   -oocmnf 5467  RR*cxr 5468   < clt 5469

This theorem is referenced by:  axlttri 5486  axlttrn 5487  axltadd 5488  axmulgt0 5489  axsup 5490

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-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  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-int 2530  df-iun 2564  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-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-enr 5149  df-nr 5150  df-0r 5154  df-c 5223  df-r 5227  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473

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