xrlttr - Metamath Proof Explorer

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Theorem xrlttr 12874

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

**Assertion**
Ref	Expression
xrlttr	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

**Proof of Theorem xrlttr**
Step	Hyp	Ref	Expression
1		elxr 12852	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 12852	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
3		elxr 12852	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4		lttr 11051	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
5	4	3expa 1117	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
6	5	an32s 649	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7		rexr 11021	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
8		pnfnlt 12864	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ^* → ¬ +∞ < 𝐶)
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℝ → ¬ +∞ < 𝐶)
10	9	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐶)
11		breq1 5077	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = +∞ → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
12	11	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
13	10, 12	mtbird 325	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐶)
14	13	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
15	14	adantll 711	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
16	15	adantld 491	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
17		rexr 11021	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
18		nltmnf 12865	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
20	19	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
21		breq2 5078	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
22	21	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
23	20, 22	mtbird 325	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
24	23	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
25	24	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
26	25	adantrd 492	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
27	6, 16, 26	3jaodan 1429	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
28	3, 27	sylan2b 594	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
29	28	an32s 649	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
30		ltpnf 12856	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
31	30	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < +∞)
32		breq2 5078	. . . . . . . . . . 11 ⊢ (𝐶 = +∞ → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
33	32	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
34	31, 33	mpbird 256	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
35	34	adantlr 712	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
36	35	a1d 25	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
37		nltmnf 12865	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
38	37	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < -∞)
39		breq2 5078	. . . . . . . . . . . 12 ⊢ (𝐶 = -∞ → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
40	39	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
41	38, 40	mtbird 325	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < 𝐶)
42	41	pm2.21d 121	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
43	42	adantld 491	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
44	43	adantll 711	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
45	29, 36, 44	3jaodan 1429	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
46	45	anasss 467	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
47		pnfnlt 12864	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
48	47	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
49		breq1 5077	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
50	49	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
51	48, 50	mtbird 325	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐴 < 𝐵)
52	51	pm2.21d 121	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
53	52	adantrd 492	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
54	53	adantrr 714	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
55		mnflt 12859	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ → -∞ < 𝐶)
56	55	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → -∞ < 𝐶)
57		breq1 5077	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
58	57	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
59	56, 58	mpbird 256	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → 𝐴 < 𝐶)
60	59	a1d 25	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
61	60	adantlr 712	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
62		mnfltpnf 12862	. . . . . . . . . 10 ⊢ -∞ < +∞
63		breq12 5079	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ -∞ < +∞))
64	62, 63	mpbiri 257	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
65	64	a1d 25	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
66	65	adantlr 712	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
67	43	adantll 711	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
68	61, 66, 67	3jaodan 1429	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
69	68	anasss 467	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
70	46, 54, 69	3jaoian 1428	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
71	70	3impb 1114	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
72	2, 71	syl3an3b 1404	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
73	1, 72	syl3an1b 1402	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ w3o 1085 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ℝcr 10870 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014

This theorem is referenced by: xrltso 12875 xrlelttr 12890 xrltletr 12891 xrlttrd 12893 xrub 13046 ioo0 13104 ioojoin 13215 hashgt23el 14139 leordtval2 22363 icopnfcld 23931 iocmnfcld 23932 ismbf3d 24818 tanord1 25693 tan2h 35769 asindmre 35860 iccpartlt 44876

W3C validator