Theorem xrlttr 10029

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 10010	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 10010	. . 3 |- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo ) )
3		elxr 10010	. . . . . . . . 9 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
4		lttr 8252	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
5	4	3expa 1229	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
6	5	an32s 570	. . . . . . . . . 10 |- ( ( ( A e. RR /\ C e. RR ) /\ B e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
7		rexr 8224	. . . . . . . . . . . . . . . 16 |- ( C e. RR -> C e. RR* )
8		pnfnlt 10021	. . . . . . . . . . . . . . . 16 |- ( C e. RR* -> -. +oo < C )
9	7, 8	syl 14	. . . . . . . . . . . . . . 15 |- ( C e. RR -> -. +oo < C )
10	9	adantr 276	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ B = +oo ) -> -. +oo < C )
11		breq1 4091	. . . . . . . . . . . . . . 15 |- ( B = +oo -> ( B < C <-> +oo < C ) )
12	11	adantl 277	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ B = +oo ) -> ( B < C <-> +oo < C ) )
13	10, 12	mtbird 679	. . . . . . . . . . . . 13 |- ( ( C e. RR /\ B = +oo ) -> -. B < C )
14	13	pm2.21d 624	. . . . . . . . . . . 12 |- ( ( C e. RR /\ B = +oo ) -> ( B < C -> A < C ) )
15	14	adantll 476	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ C e. RR ) /\ B = +oo ) -> ( B < C -> A < C ) )
16	15	adantld 278	. . . . . . . . . 10 |- ( ( ( A e. RR /\ C e. RR ) /\ B = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
17		rexr 8224	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> A e. RR* )
18		nltmnf 10022	. . . . . . . . . . . . . . . 16 |- ( A e. RR* -> -. A < -oo )
19	17, 18	syl 14	. . . . . . . . . . . . . . 15 |- ( A e. RR -> -. A < -oo )
20	19	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
21		breq2 4092	. . . . . . . . . . . . . . 15 |- ( B = -oo -> ( A < B <-> A < -oo ) )
22	21	adantl 277	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
23	20, 22	mtbird 679	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
24	23	pm2.21d 624	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> A < C ) )
25	24	adantlr 477	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ C e. RR ) /\ B = -oo ) -> ( A < B -> A < C ) )
26	25	adantrd 279	. . . . . . . . . 10 |- ( ( ( A e. RR /\ C e. RR ) /\ B = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
27	6, 16, 26	3jaodan 1342	. . . . . . . . 9 |- ( ( ( A e. RR /\ C e. RR ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
28	3, 27	sylan2b 287	. . . . . . . 8 |- ( ( ( A e. RR /\ C e. RR ) /\ B e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
29	28	an32s 570	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
30		ltpnf 10014	. . . . . . . . . . 11 |- ( A e. RR -> A < +oo )
31	30	adantr 276	. . . . . . . . . 10 |- ( ( A e. RR /\ C = +oo ) -> A < +oo )
32		breq2 4092	. . . . . . . . . . 11 |- ( C = +oo -> ( A < C <-> A < +oo ) )
33	32	adantl 277	. . . . . . . . . 10 |- ( ( A e. RR /\ C = +oo ) -> ( A < C <-> A < +oo ) )
34	31, 33	mpbird 167	. . . . . . . . 9 |- ( ( A e. RR /\ C = +oo ) -> A < C )
35	34	adantlr 477	. . . . . . . 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 10022	. . . . . . . . . . . 12 |- ( B e. RR* -> -. B < -oo )
38	37	adantr 276	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C = -oo ) -> -. B < -oo )
39		breq2 4092	. . . . . . . . . . . 12 |- ( C = -oo -> ( B < C <-> B < -oo ) )
40	39	adantl 277	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C = -oo ) -> ( B < C <-> B < -oo ) )
41	38, 40	mtbird 679	. . . . . . . . . 10 |- ( ( B e. RR* /\ C = -oo ) -> -. B < C )
42	41	pm2.21d 624	. . . . . . . . 9 |- ( ( B e. RR* /\ C = -oo ) -> ( B < C -> A < C ) )
43	42	adantld 278	. . . . . . . 8 |- ( ( B e. RR* /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
44	43	adantll 476	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
45	29, 36, 44	3jaodan 1342	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
46	45	anasss 399	. . . . 5 |- ( ( A e. RR /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
47		pnfnlt 10021	. . . . . . . . . 10 |- ( B e. RR* -> -. +oo < B )
48	47	adantl 277	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
49		breq1 4091	. . . . . . . . . 10 |- ( A = +oo -> ( A < B <-> +oo < B ) )
50	49	adantr 276	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
51	48, 50	mtbird 679	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
52	51	pm2.21d 624	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> A < C ) )
53	52	adantrd 279	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
54	53	adantrr 479	. . . . 5 |- ( ( A = +oo /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
55		mnflt 10017	. . . . . . . . . . 11 |- ( C e. RR -> -oo < C )
56	55	adantl 277	. . . . . . . . . 10 |- ( ( A = -oo /\ C e. RR ) -> -oo < C )
57		breq1 4091	. . . . . . . . . . 11 |- ( A = -oo -> ( A < C <-> -oo < C ) )
58	57	adantr 276	. . . . . . . . . 10 |- ( ( A = -oo /\ C e. RR ) -> ( A < C <-> -oo < C ) )
59	56, 58	mpbird 167	. . . . . . . . 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 477	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
62		mnfltpnf 10019	. . . . . . . . . 10 |- -oo < +oo
63		breq12 4093	. . . . . . . . . 10 |- ( ( A = -oo /\ C = +oo ) -> ( A < C <-> -oo < +oo ) )
64	62, 63	mpbiri 168	. . . . . . . . 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 477	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ C = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
67	43	adantll 476	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
68	61, 66, 67	3jaodan 1342	. . . . . 6 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
69	68	anasss 399	. . . . 5 |- ( ( A = -oo /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
70	46, 54, 69	3jaoian 1341	. . . 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 1225	. . 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 1311	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
73	1, 72	syl3an1b 1309	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1003   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   RRcr 8030   +oocpnf 8210   -oocmnf 8211   RR*cxr 8212   < clt 8213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-lttrn 8145

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218

This theorem is referenced by:  xrltso  10030  xrlttrd  10043  ioo0  10518