Theorem xrlttr 6728

Description: Ordering on the extended reals is transitive.

Assertion

Ref	Expression
xrlttr	|- ((A e. RR* /\ B e. RR* /\ C e. RR*) -> ((A < B /\ B < C) -> A < C))

Proof of Theorem xrlttr

Step	Hyp	Ref	Expression
1		axlttrn 6673	. . . . . . . . . . . 12 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
2	1	3expa 1067	. . . . . . . . . . 11 |- (((A e. RR /\ B e. RR) /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
3	2	an1rs 547	. . . . . . . . . 10 |- (((A e. RR /\ C e. RR) /\ B e. RR) -> ((A < B /\ B < C) -> A < C))
4		rexr 6668	. . . . . . . . . . . . . . . 16 |- (C e. RR -> C e. RR*)
5		pnfnlt 6721	. . . . . . . . . . . . . . . 16 |- (C e. RR* -> -. +oo < C)
6	4, 5	syl 12	. . . . . . . . . . . . . . 15 |- (C e. RR -> -. +oo < C)
7	6	adantr 425	. . . . . . . . . . . . . 14 |- ((C e. RR /\ B = +oo) -> -. +oo < C)
8		breq1 3341	. . . . . . . . . . . . . . 15 |- (B = +oo -> (B < C <-> +oo < C))
9	8	adantl 424	. . . . . . . . . . . . . 14 |- ((C e. RR /\ B = +oo) -> (B < C <-> +oo < C))
10	7, 9	mtbird 783	. . . . . . . . . . . . 13 |- ((C e. RR /\ B = +oo) -> -. B < C)
11	10	pm2.21d 94	. . . . . . . . . . . 12 |- ((C e. RR /\ B = +oo) -> (B < C -> A < C))
12	11	adantll 428	. . . . . . . . . . 11 |- (((A e. RR /\ C e. RR) /\ B = +oo) -> (B < C -> A < C))
13	12	adantld 426	. . . . . . . . . 10 |- (((A e. RR /\ C e. RR) /\ B = +oo) -> ((A < B /\ B < C) -> A < C))
14		rexr 6668	. . . . . . . . . . . . . . . 16 |- (A e. RR -> A e. RR*)
15		nltmnf 6722	. . . . . . . . . . . . . . . 16 |- (A e. RR* -> -. A < -oo)
16	14, 15	syl 12	. . . . . . . . . . . . . . 15 |- (A e. RR -> -. A < -oo)
17	16	adantr 425	. . . . . . . . . . . . . 14 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
18		breq2 3342	. . . . . . . . . . . . . . 15 |- (B = -oo -> (A < B <-> A < -oo))
19	18	adantl 424	. . . . . . . . . . . . . 14 |- ((A e. RR /\ B = -oo) -> (A < B <-> A < -oo))
20	17, 19	mtbird 783	. . . . . . . . . . . . 13 |- ((A e. RR /\ B = -oo) -> -. A < B)
21	20	pm2.21d 94	. . . . . . . . . . . 12 |- ((A e. RR /\ B = -oo) -> (A < B -> A < C))
22	21	adantlr 429	. . . . . . . . . . 11 |- (((A e. RR /\ C e. RR) /\ B = -oo) -> (A < B -> A < C))
23	22	adantrd 427	. . . . . . . . . 10 |- (((A e. RR /\ C e. RR) /\ B = -oo) -> ((A < B /\ B < C) -> A < C))
24	3, 13, 23	3jaodan 1163	. . . . . . . . 9 |- (((A e. RR /\ C e. RR) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> ((A < B /\ B < C) -> A < C))
25		elxr 6706	. . . . . . . . 9 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
26	24, 25	sylan2b 501	. . . . . . . 8 |- (((A e. RR /\ C e. RR) /\ B e. RR*) -> ((A < B /\ B < C) -> A < C))
27	26	an1rs 547	. . . . . . 7 |- (((A e. RR /\ B e. RR*) /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
28		ltpnf 6717	. . . . . . . . . . 11 |- (A e. RR -> A < +oo)
29	28	adantr 425	. . . . . . . . . 10 |- ((A e. RR /\ C = +oo) -> A < +oo)
30		breq2 3342	. . . . . . . . . . 11 |- (C = +oo -> (A < C <-> A < +oo))
31	30	adantl 424	. . . . . . . . . 10 |- ((A e. RR /\ C = +oo) -> (A < C <-> A < +oo))
32	29, 31	mpbird 213	. . . . . . . . 9 |- ((A e. RR /\ C = +oo) -> A < C)
33	32	adantlr 429	. . . . . . . 8 |- (((A e. RR /\ B e. RR*) /\ C = +oo) -> A < C)
34	33	a1d 15	. . . . . . 7 |- (((A e. RR /\ B e. RR*) /\ C = +oo) -> ((A < B /\ B < C) -> A < C))
35		nltmnf 6722	. . . . . . . . . . . 12 |- (B e. RR* -> -. B < -oo)
36	35	adantr 425	. . . . . . . . . . 11 |- ((B e. RR* /\ C = -oo) -> -. B < -oo)
37		breq2 3342	. . . . . . . . . . . 12 |- (C = -oo -> (B < C <-> B < -oo))
38	37	adantl 424	. . . . . . . . . . 11 |- ((B e. RR* /\ C = -oo) -> (B < C <-> B < -oo))
39	36, 38	mtbird 783	. . . . . . . . . 10 |- ((B e. RR* /\ C = -oo) -> -. B < C)
40	39	pm2.21d 94	. . . . . . . . 9 |- ((B e. RR* /\ C = -oo) -> (B < C -> A < C))
41	40	adantld 426	. . . . . . . 8 |- ((B e. RR* /\ C = -oo) -> ((A < B /\ B < C) -> A < C))
42	41	adantll 428	. . . . . . 7 |- (((A e. RR /\ B e. RR*) /\ C = -oo) -> ((A < B /\ B < C) -> A < C))
43	27, 34, 42	3jaodan 1163	. . . . . 6 |- (((A e. RR /\ B e. RR*) /\ (C e. RR \/ C = +oo \/ C = -oo)) -> ((A < B /\ B < C) -> A < C))
44	43	anasss 488	. . . . 5 |- ((A e. RR /\ (B e. RR* /\ (C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
45		pnfnlt 6721	. . . . . . . . . 10 |- (B e. RR* -> -. +oo < B)
46	45	adantl 424	. . . . . . . . 9 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
47		breq1 3341	. . . . . . . . . 10 |- (A = +oo -> (A < B <-> +oo < B))
48	47	adantr 425	. . . . . . . . 9 |- ((A = +oo /\ B e. RR*) -> (A < B <-> +oo < B))
49	46, 48	mtbird 783	. . . . . . . 8 |- ((A = +oo /\ B e. RR*) -> -. A < B)
50	49	pm2.21d 94	. . . . . . 7 |- ((A = +oo /\ B e. RR*) -> (A < B -> A < C))
51	50	adantrd 427	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> ((A < B /\ B < C) -> A < C))
52	51	adantrr 431	. . . . 5 |- ((A = +oo /\ (B e. RR* /\ (C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
53		mnflt 6718	. . . . . . . . . . 11 |- (C e. RR -> -oo < C)
54	53	adantl 424	. . . . . . . . . 10 |- ((A = -oo /\ C e. RR) -> -oo < C)
55		breq1 3341	. . . . . . . . . . 11 |- (A = -oo -> (A < C <-> -oo < C))
56	55	adantr 425	. . . . . . . . . 10 |- ((A = -oo /\ C e. RR) -> (A < C <-> -oo < C))
57	54, 56	mpbird 213	. . . . . . . . 9 |- ((A = -oo /\ C e. RR) -> A < C)
58	57	a1d 15	. . . . . . . 8 |- ((A = -oo /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
59	58	adantlr 429	. . . . . . 7 |- (((A = -oo /\ B e. RR*) /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
60		mnfltpnf 6719	. . . . . . . . . 10 |- -oo < +oo
61		breq12 3343	. . . . . . . . . 10 |- ((A = -oo /\ C = +oo) -> (A < C <-> -oo < +oo))
62	60, 61	mpbiri 211	. . . . . . . . 9 |- ((A = -oo /\ C = +oo) -> A < C)
63	62	a1d 15	. . . . . . . 8 |- ((A = -oo /\ C = +oo) -> ((A < B /\ B < C) -> A < C))
64	63	adantlr 429	. . . . . . 7 |- (((A = -oo /\ B e. RR*) /\ C = +oo) -> ((A < B /\ B < C) -> A < C))
65	41	adantll 428	. . . . . . 7 |- (((A = -oo /\ B e. RR*) /\ C = -oo) -> ((A < B /\ B < C) -> A < C))
66	59, 64, 65	3jaodan 1163	. . . . . 6 |- (((A = -oo /\ B e. RR*) /\ (C e. RR \/ C = +oo \/ C = -oo)) -> ((A < B /\ B < C) -> A < C))
67	66	anasss 488	. . . . 5 |- ((A = -oo /\ (B e. RR* /\ (C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
68	44, 52, 67	3jaoian 1162	. . . 4 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR* /\ (C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
69	68	3impb 1063	. . 3 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ B e. RR* /\ (C e. RR \/ C = +oo \/ C = -oo)) -> ((A < B /\ B < C) -> A < C))
70		elxr 6706	. . 3 |- (C e. RR* <-> (C e. RR \/ C = +oo \/ C = -oo))
71	69, 70	syl3an3b 1135	. 2 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ B e. RR* /\ C e. RR*) -> ((A < B /\ B < C) -> A < C))
72		elxr 6706	. 2 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
73	71, 72	syl3an1b 1133	1 |- ((A e. RR* /\ B e. RR* /\ C e. RR*) -> ((A < B /\ B < C) -> A < C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  RRcr 6385   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652   < clt 6653

This theorem is referenced by:  xrltso 6729  xrlelttr 6737  xrltletr 6738  xrub 7289  qbtwnxr 7460  ioo0 7535  elioc2 7558  elico2 7559  elioo1t3 14846  truni1 14849  truni3 14851  ioodisj 15364

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-plp 6240  df-ltp 6242  df-enr 6318  df-nr 6319  df-ltr 6322  df-0r 6323  df-c 6392  df-r 6396  df-lt 6399  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657

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