Theorem ltxrlt 6669

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
ltxrlt	|- ((A e. RR /\ B e. RR) -> (A < B <-> A <R B))

Proof of Theorem ltxrlt

Step	Hyp	Ref	Expression
1		ltxr 6664	. . 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		rexr 6668	. . 3 |- (A e. RR -> A e. RR*)
3		rexr 6668	. . 3 |- (B e. RR -> B e. RR*)
4	1, 2, 3	syl2an 503	. 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 705	. . . 4 |- ((A e. RR /\ B e. RR) -> (A <R B <-> ((A e. RR /\ B e. RR) /\ A <R B)))
6		ioran 331	. . . . . 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))))
7		pnfnre 6665	. . . . . . . . . . 11 |- +oo e/ RR
8		df-nel 2020	. . . . . . . . . . 11 |- ( +oo e/ RR <-> -. +oo e. RR)
9	7, 8	mpbi 206	. . . . . . . . . 10 |- -. +oo e. RR
10		eleq1 1957	. . . . . . . . . 10 |- (B = +oo -> (B e. RR <-> +oo e. RR))
11	9, 10	mtbiri 785	. . . . . . . . 9 |- (B = +oo -> -. B e. RR)
12	11	con2i 113	. . . . . . . 8 |- (B e. RR -> -. B = +oo)
13	12	intnand 757	. . . . . . 7 |- (B e. RR -> -. (A = -oo /\ B = +oo))
14	13	adantl 424	. . . . . 6 |- ((A e. RR /\ B e. RR) -> -. (A = -oo /\ B = +oo))
15	12	intnand 757	. . . . . . . . 9 |- (B e. RR -> -. (A e. RR /\ B = +oo))
16		mnfnre 6666	. . . . . . . . . . . . 13 |- -oo e/ RR
17		df-nel 2020	. . . . . . . . . . . . 13 |- ( -oo e/ RR <-> -. -oo e. RR)
18	16, 17	mpbi 206	. . . . . . . . . . . 12 |- -. -oo e. RR
19		eleq1 1957	. . . . . . . . . . . 12 |- (A = -oo -> (A e. RR <-> -oo e. RR))
20	18, 19	mtbiri 785	. . . . . . . . . . 11 |- (A = -oo -> -. A e. RR)
21	20	con2i 113	. . . . . . . . . 10 |- (A e. RR -> -. A = -oo)
22	21	intnanrd 758	. . . . . . . . 9 |- (A e. RR -> -. (A = -oo /\ B e. RR))
23	15, 22	anim12i 360	. . . . . . . 8 |- ((B e. RR /\ A e. RR) -> (-. (A e. RR /\ B = +oo) /\ -. (A = -oo /\ B e. RR)))
24		ioran 331	. . . . . . . 8 |- (-. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)) <-> (-. (A e. RR /\ B = +oo) /\ -. (A = -oo /\ B e. RR)))
25	23, 24	sylibr 217	. . . . . . 7 |- ((B e. RR /\ A e. RR) -> -. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))
26	25	ancoms 484	. . . . . 6 |- ((A e. RR /\ B e. RR) -> -. ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR)))
27	6, 14, 26	sylanbrc 527	. . . . 5 |- ((A e. RR /\ B e. RR) -> -. ((A = -oo /\ B = +oo) \/ ((A e. RR /\ B = +oo) \/ (A = -oo /\ B e. RR))))
28		biorf 807	. . . . 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))))
29	27, 28	syl 12	. . . 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))))
30	5, 29	bitr2d 588	. . 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))
31		orass 280	. . . 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)))))
32		orcom 266	. . . 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)))
33	31, 32	bitri 190	. . 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)))
34	30, 33	syl5bb 591	. 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))
35	4, 34	bitrd 587	1 |- ((A e. RR /\ B e. RR) -> (A < B <-> A <R B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   e/ wnel 2018   class class class wbr 3338  RRcr 6385   <R cltrr 6390   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652   < clt 6653

This theorem is referenced by:  axlttri 6672  axlttrn 6673  axltadd 6674  axmulgt0 6675  axsup 6676

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-enr 6318  df-nr 6319  df-0r 6323  df-c 6392  df-r 6396  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657

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