Theorem xrlttr 9731

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 9712	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		elxr 9712	. . 3 |- ( C e. RR* <-> ( C e. RR \/ C = +oo \/ C = -oo ) )
3		elxr 9712	. . . . . . . . 9 |- ( B e. RR* <-> ( B e. RR \/ B = +oo \/ B = -oo ) )
4		lttr 7972	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
5	4	3expa 1193	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
6	5	an32s 558	. . . . . . . . . 10 |- ( ( ( A e. RR /\ C e. RR ) /\ B e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
7		rexr 7944	. . . . . . . . . . . . . . . 16 |- ( C e. RR -> C e. RR* )
8		pnfnlt 9723	. . . . . . . . . . . . . . . 16 |- ( C e. RR* -> -. +oo < C )
9	7, 8	syl 14	. . . . . . . . . . . . . . 15 |- ( C e. RR -> -. +oo < C )
10	9	adantr 274	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ B = +oo ) -> -. +oo < C )
11		breq1 3985	. . . . . . . . . . . . . . 15 |- ( B = +oo -> ( B < C <-> +oo < C ) )
12	11	adantl 275	. . . . . . . . . . . . . 14 |- ( ( C e. RR /\ B = +oo ) -> ( B < C <-> +oo < C ) )
13	10, 12	mtbird 663	. . . . . . . . . . . . 13 |- ( ( C e. RR /\ B = +oo ) -> -. B < C )
14	13	pm2.21d 609	. . . . . . . . . . . 12 |- ( ( C e. RR /\ B = +oo ) -> ( B < C -> A < C ) )
15	14	adantll 468	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ C e. RR ) /\ B = +oo ) -> ( B < C -> A < C ) )
16	15	adantld 276	. . . . . . . . . 10 |- ( ( ( A e. RR /\ C e. RR ) /\ B = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
17		rexr 7944	. . . . . . . . . . . . . . . 16 |- ( A e. RR -> A e. RR* )
18		nltmnf 9724	. . . . . . . . . . . . . . . 16 |- ( A e. RR* -> -. A < -oo )
19	17, 18	syl 14	. . . . . . . . . . . . . . 15 |- ( A e. RR -> -. A < -oo )
20	19	adantr 274	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B = -oo ) -> -. A < -oo )
21		breq2 3986	. . . . . . . . . . . . . . 15 |- ( B = -oo -> ( A < B <-> A < -oo ) )
22	21	adantl 275	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B = -oo ) -> ( A < B <-> A < -oo ) )
23	20, 22	mtbird 663	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B = -oo ) -> -. A < B )
24	23	pm2.21d 609	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B = -oo ) -> ( A < B -> A < C ) )
25	24	adantlr 469	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ C e. RR ) /\ B = -oo ) -> ( A < B -> A < C ) )
26	25	adantrd 277	. . . . . . . . . 10 |- ( ( ( A e. RR /\ C e. RR ) /\ B = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
27	6, 16, 26	3jaodan 1296	. . . . . . . . 9 |- ( ( ( A e. RR /\ C e. RR ) /\ ( B e. RR \/ B = +oo \/ B = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
28	3, 27	sylan2b 285	. . . . . . . 8 |- ( ( ( A e. RR /\ C e. RR ) /\ B e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
29	28	an32s 558	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
30		ltpnf 9716	. . . . . . . . . . 11 |- ( A e. RR -> A < +oo )
31	30	adantr 274	. . . . . . . . . 10 |- ( ( A e. RR /\ C = +oo ) -> A < +oo )
32		breq2 3986	. . . . . . . . . . 11 |- ( C = +oo -> ( A < C <-> A < +oo ) )
33	32	adantl 275	. . . . . . . . . 10 |- ( ( A e. RR /\ C = +oo ) -> ( A < C <-> A < +oo ) )
34	31, 33	mpbird 166	. . . . . . . . 9 |- ( ( A e. RR /\ C = +oo ) -> A < C )
35	34	adantlr 469	. . . . . . . 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 9724	. . . . . . . . . . . 12 |- ( B e. RR* -> -. B < -oo )
38	37	adantr 274	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C = -oo ) -> -. B < -oo )
39		breq2 3986	. . . . . . . . . . . 12 |- ( C = -oo -> ( B < C <-> B < -oo ) )
40	39	adantl 275	. . . . . . . . . . 11 |- ( ( B e. RR* /\ C = -oo ) -> ( B < C <-> B < -oo ) )
41	38, 40	mtbird 663	. . . . . . . . . 10 |- ( ( B e. RR* /\ C = -oo ) -> -. B < C )
42	41	pm2.21d 609	. . . . . . . . 9 |- ( ( B e. RR* /\ C = -oo ) -> ( B < C -> A < C ) )
43	42	adantld 276	. . . . . . . 8 |- ( ( B e. RR* /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
44	43	adantll 468	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR* ) /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
45	29, 36, 44	3jaodan 1296	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR* ) /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
46	45	anasss 397	. . . . 5 |- ( ( A e. RR /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
47		pnfnlt 9723	. . . . . . . . . 10 |- ( B e. RR* -> -. +oo < B )
48	47	adantl 275	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR* ) -> -. +oo < B )
49		breq1 3985	. . . . . . . . . 10 |- ( A = +oo -> ( A < B <-> +oo < B ) )
50	49	adantr 274	. . . . . . . . 9 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B <-> +oo < B ) )
51	48, 50	mtbird 663	. . . . . . . 8 |- ( ( A = +oo /\ B e. RR* ) -> -. A < B )
52	51	pm2.21d 609	. . . . . . 7 |- ( ( A = +oo /\ B e. RR* ) -> ( A < B -> A < C ) )
53	52	adantrd 277	. . . . . 6 |- ( ( A = +oo /\ B e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
54	53	adantrr 471	. . . . 5 |- ( ( A = +oo /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
55		mnflt 9719	. . . . . . . . . . 11 |- ( C e. RR -> -oo < C )
56	55	adantl 275	. . . . . . . . . 10 |- ( ( A = -oo /\ C e. RR ) -> -oo < C )
57		breq1 3985	. . . . . . . . . . 11 |- ( A = -oo -> ( A < C <-> -oo < C ) )
58	57	adantr 274	. . . . . . . . . 10 |- ( ( A = -oo /\ C e. RR ) -> ( A < C <-> -oo < C ) )
59	56, 58	mpbird 166	. . . . . . . . 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 469	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
62		mnfltpnf 9721	. . . . . . . . . 10 |- -oo < +oo
63		breq12 3987	. . . . . . . . . 10 |- ( ( A = -oo /\ C = +oo ) -> ( A < C <-> -oo < +oo ) )
64	62, 63	mpbiri 167	. . . . . . . . 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 469	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ C = +oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
67	43	adantll 468	. . . . . . 7 |- ( ( ( A = -oo /\ B e. RR* ) /\ C = -oo ) -> ( ( A < B /\ B < C ) -> A < C ) )
68	61, 66, 67	3jaodan 1296	. . . . . 6 |- ( ( ( A = -oo /\ B e. RR* ) /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
69	68	anasss 397	. . . . 5 |- ( ( A = -oo /\ ( B e. RR* /\ ( C e. RR \/ C = +oo \/ C = -oo ) ) ) -> ( ( A < B /\ B < C ) -> A < C ) )
70	46, 54, 69	3jaoian 1295	. . . 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 1189	. . 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 1266	. 2 |- ( ( ( A e. RR \/ A = +oo \/ A = -oo ) /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
73	1, 72	syl3an1b 1264	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 967   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-lttrn 7867

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938

This theorem is referenced by:  xrltso  9732  xrlttrd  9745  ioo0  10195