Theorem ltxrltt 5472

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 5467	. . 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 5471	. . 3 |- (A e. RR -> A e. RR*)
3		rexrt 5471	. . 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 641	. . . 4 |- ((A e. RR /\ B e. RR) -> (A <R B <-> ((A e. RR /\ B e. RR) /\ A <R B)))
6		pnfnre 5468	. . . . . . . . . . . 12 |- +oo e/ RR
7		df-nel 1580	. . . . . . . . . . . 12 |- ( +oo e/ RR <-> -. +oo e. RR)
8	6, 7	mpbi 189	. . . . . . . . . . 11 |- -. +oo e. RR
9		eleq1 1526	. . . . . . . . . . 11 |- (B = +oo -> (B e. RR <-> +oo e. RR))
10	8, 9	mtbiri 715	. . . . . . . . . 10 |- (B = +oo -> -. B e. RR)
11	10	con2i 97	. . . . . . . . 9 |- (B e. RR -> -. B = +oo)
12	11	intnand 691	. . . . . . . 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 691	. . . . . . . . . 10 |- (B e. RR -> -. (A e. RR /\ B = +oo))
15		mnfnre 5469	. . . . . . . . . . . . . 14 |- -oo e/ RR
16		df-nel 1580	. . . . . . . . . . . . . 14 |- ( -oo e/ RR <-> -. -oo e. RR)
17	15, 16	mpbi 189	. . . . . . . . . . . . 13 |- -. -oo e. RR
18		eleq1 1526	. . . . . . . . . . . . 13 |- (A = -oo -> (A e. RR <-> -oo e. RR))
19	17, 18	mtbiri 715	. . . . . . . . . . . 12 |- (A = -oo -> -. A e. RR)
20	19	con2i 97	. . . . . . . . . . 11 |- (A e. RR -> -. A = -oo)
21	20	intnanrd 692	. . . . . . . . . 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 733	. . . . 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 527	. . 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 530	. 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 526	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 953   e. wcel 955   e/ wnel 1578   class class class wbr 2609  RRcr 5205   <R cltrr 5210   +oocpnf 5455   -oocmnf 5456  RR*cxr 5457   < clt 5458

This theorem is referenced by:  axlttri 5475  axlttrn 5476  axltadd 5477  axmulgt0 5478  axsup 5479

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-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  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-int 2524  df-iun 2558  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-enr 5138  df-nr 5139  df-0r 5143  df-c 5212  df-r 5216  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462

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