Theorem xrlttr 8486

Description: Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)

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		elxr 8466	. 2 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
2		elxr 8466	. . 3 |- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo))
3		elxr 8466	. . . . . . . . 9 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
4		lttr 6889	. . . . . . . . . . . 12 |- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
5	4	3expa 1103	. . . . . . . . . . 11 |- (((A e. RR /\ B e. RR) /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
6	5	an32s 502	. . . . . . . . . 10 |- (((A e. RR /\ C e. RR) /\ B e. RR) -> ((A < B /\ B < C) -> A < C))
7		rexr 6868	. . . . . . . . . . . . . . . 16 |- ( C e. RR -> C e. RR*)
8		pnfnlt 8478	. . . . . . . . . . . . . . . 16 |- ( C e. RR* -> -. +oo < C)
9	7, 8	syl 14	. . . . . . . . . . . . . . 15 |- ( C e. RR -> -. +oo < C)
10	9	adantr 261	. . . . . . . . . . . . . 14 |- (( C e. RR /\ B = +oo) -> -. +oo < C)
11		breq1 3758	. . . . . . . . . . . . . . 15 |- (B = +oo -> (B < C <-> +oo < C))
12	11	adantl 262	. . . . . . . . . . . . . 14 |- (( C e. RR /\ B = +oo) -> (B < C <-> +oo < C))
13	10, 12	mtbird 597	. . . . . . . . . . . . 13 |- (( C e. RR /\ B = +oo) -> -. B < C)
14	13	pm2.21d 549	. . . . . . . . . . . 12 |- (( C e. RR /\ B = +oo) -> (B < C -> A < C))
15	14	adantll 445	. . . . . . . . . . 11 |- (((A e. RR /\ C e. RR) /\ B = +oo) -> (B < C -> A < C))
16	15	adantld 263	. . . . . . . . . 10 |- (((A e. RR /\ C e. RR) /\ B = +oo) -> ((A < B /\ B < C) -> A < C))
17		rexr 6868	. . . . . . . . . . . . . . . 16 |- (A e. RR -> A e. RR*)
18		nltmnf 8479	. . . . . . . . . . . . . . . 16 |- (A e. RR* -> -. A < -oo)
19	17, 18	syl 14	. . . . . . . . . . . . . . 15 |- (A e. RR -> -. A < -oo)
20	19	adantr 261	. . . . . . . . . . . . . 14 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
21		breq2 3759	. . . . . . . . . . . . . . 15 |- (B = -oo -> (A < B <-> A < -oo))
22	21	adantl 262	. . . . . . . . . . . . . 14 |- ((A e. RR /\ B = -oo) -> (A < B <-> A < -oo))
23	20, 22	mtbird 597	. . . . . . . . . . . . 13 |- ((A e. RR /\ B = -oo) -> -. A < B)
24	23	pm2.21d 549	. . . . . . . . . . . 12 |- ((A e. RR /\ B = -oo) -> (A < B -> A < C))
25	24	adantlr 446	. . . . . . . . . . 11 |- (((A e. RR /\ C e. RR) /\ B = -oo) -> (A < B -> A < C))
26	25	adantrd 264	. . . . . . . . . 10 |- (((A e. RR /\ C e. RR) /\ B = -oo) -> ((A < B /\ B < C) -> A < C))
27	6, 16, 26	3jaodan 1200	. . . . . . . . 9 |- (((A e. RR /\ C e. RR) /\ (B e. RR \/ B = +oo \/ B = -oo)) -> ((A < B /\ B < C) -> A < C))
28	3, 27	sylan2b 271	. . . . . . . 8 |- (((A e. RR /\ C e. RR) /\ B e. RR*) -> ((A < B /\ B < C) -> A < C))
29	28	an32s 502	. . . . . . 7 |- (((A e. RR /\ B e. RR*) /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
30		ltpnf 8472	. . . . . . . . . . 11 |- (A e. RR -> A < +oo)
31	30	adantr 261	. . . . . . . . . 10 |- ((A e. RR /\ C = +oo) -> A < +oo)
32		breq2 3759	. . . . . . . . . . 11 |- ( C = +oo -> (A < C <-> A < +oo))
33	32	adantl 262	. . . . . . . . . 10 |- ((A e. RR /\ C = +oo) -> (A < C <-> A < +oo))
34	31, 33	mpbird 156	. . . . . . . . 9 |- ((A e. RR /\ C = +oo) -> A < C)
35	34	adantlr 446	. . . . . . . 8 |- (((A e. RR /\ B e. RR*) /\ C = +oo) -> A < C)
36	35	a1d 22	. . . . . . 7 |- (((A e. RR /\ B e. RR*) /\ C = +oo) -> ((A < B /\ B < C) -> A < C))
37		nltmnf 8479	. . . . . . . . . . . 12 |- (B e. RR* -> -. B < -oo)
38	37	adantr 261	. . . . . . . . . . 11 |- ((B e. RR* /\ C = -oo) -> -. B < -oo)
39		breq2 3759	. . . . . . . . . . . 12 |- ( C = -oo -> (B < C <-> B < -oo))
40	39	adantl 262	. . . . . . . . . . 11 |- ((B e. RR* /\ C = -oo) -> (B < C <-> B < -oo))
41	38, 40	mtbird 597	. . . . . . . . . 10 |- ((B e. RR* /\ C = -oo) -> -. B < C)
42	41	pm2.21d 549	. . . . . . . . 9 |- ((B e. RR* /\ C = -oo) -> (B < C -> A < C))
43	42	adantld 263	. . . . . . . 8 |- ((B e. RR* /\ C = -oo) -> ((A < B /\ B < C) -> A < C))
44	43	adantll 445	. . . . . . 7 |- (((A e. RR /\ B e. RR*) /\ C = -oo) -> ((A < B /\ B < C) -> A < C))
45	29, 36, 44	3jaodan 1200	. . . . . 6 |- (((A e. RR /\ B e. RR*) /\ ( C e. RR \/ C = +oo \/ C = -oo)) -> ((A < B /\ B < C) -> A < C))
46	45	anasss 379	. . . . 5 |- ((A e. RR /\ (B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
47		pnfnlt 8478	. . . . . . . . . 10 |- (B e. RR* -> -. +oo < B)
48	47	adantl 262	. . . . . . . . 9 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
49		breq1 3758	. . . . . . . . . 10 |- (A = +oo -> (A < B <-> +oo < B))
50	49	adantr 261	. . . . . . . . 9 |- ((A = +oo /\ B e. RR*) -> (A < B <-> +oo < B))
51	48, 50	mtbird 597	. . . . . . . 8 |- ((A = +oo /\ B e. RR*) -> -. A < B)
52	51	pm2.21d 549	. . . . . . 7 |- ((A = +oo /\ B e. RR*) -> (A < B -> A < C))
53	52	adantrd 264	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> ((A < B /\ B < C) -> A < C))
54	53	adantrr 448	. . . . 5 |- ((A = +oo /\ (B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
55		mnflt 8474	. . . . . . . . . . 11 |- ( C e. RR -> -oo < C)
56	55	adantl 262	. . . . . . . . . 10 |- ((A = -oo /\ C e. RR) -> -oo < C)
57		breq1 3758	. . . . . . . . . . 11 |- (A = -oo -> (A < C <-> -oo < C))
58	57	adantr 261	. . . . . . . . . 10 |- ((A = -oo /\ C e. RR) -> (A < C <-> -oo < C))
59	56, 58	mpbird 156	. . . . . . . . 9 |- ((A = -oo /\ C e. RR) -> A < C)
60	59	a1d 22	. . . . . . . 8 |- ((A = -oo /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
61	60	adantlr 446	. . . . . . 7 |- (((A = -oo /\ B e. RR*) /\ C e. RR) -> ((A < B /\ B < C) -> A < C))
62		mnfltpnf 8476	. . . . . . . . . 10 |- -oo < +oo
63		breq12 3760	. . . . . . . . . 10 |- ((A = -oo /\ C = +oo) -> (A < C <-> -oo < +oo))
64	62, 63	mpbiri 157	. . . . . . . . 9 |- ((A = -oo /\ C = +oo) -> A < C)
65	64	a1d 22	. . . . . . . 8 |- ((A = -oo /\ C = +oo) -> ((A < B /\ B < C) -> A < C))
66	65	adantlr 446	. . . . . . 7 |- (((A = -oo /\ B e. RR*) /\ C = +oo) -> ((A < B /\ B < C) -> A < C))
67	43	adantll 445	. . . . . . 7 |- (((A = -oo /\ B e. RR*) /\ C = -oo) -> ((A < B /\ B < C) -> A < C))
68	61, 66, 67	3jaodan 1200	. . . . . 6 |- (((A = -oo /\ B e. RR*) /\ ( C e. RR \/ C = +oo \/ C = -oo)) -> ((A < B /\ B < C) -> A < C))
69	68	anasss 379	. . . . 5 |- ((A = -oo /\ (B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
70	46, 54, 69	3jaoian 1199	. . . 4 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ (B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo))) -> ((A < B /\ B < C) -> A < C))
71	70	3impb 1099	. . 3 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo)) -> ((A < B /\ B < C) -> A < C))
72	2, 71	syl3an3b 1172	. 2 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ B e. RR* /\ C e. RR*) -> ((A < B /\ B < C) -> A < C))
73	1, 72	syl3an1b 1170	1 |- ((A e. RR* /\ B e. RR* /\ C e. RR*) -> ((A < B /\ B < C) -> A < C))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ w3o 883   /\ w3a 884   = wceq 1242   e. wcel 1390   class class class wbr 3755   RRcr 6710   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856   < clt 6857

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-lttrn 6797

This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862

This theorem is referenced by:  xrltso  8487  xrlttrd  8495