xrlttr - Metamath Proof Explorer

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Theorem xrlttr 13161

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 13137	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13137	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
3		elxr 13137	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4		lttr 11316	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
5	4	3expa 1118	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
6	5	an32s 652	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7		rexr 11286	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
8		pnfnlt 13149	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ^* → ¬ +∞ < 𝐶)
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℝ → ¬ +∞ < 𝐶)
10	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐶)
11		breq1 5127	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = +∞ → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
12	11	adantl 481	. . . . . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
16	15	adantld 490	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
17		rexr 11286	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
18		nltmnf 13150	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
21		breq2 5128	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
22	21	adantl 481	. . . . . . . . . . . . . 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 715	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
26	25	adantrd 491	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
27	6, 16, 26	3jaodan 1433	. . . . . . . . 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 652	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
30		ltpnf 13141	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < +∞)
32		breq2 5128	. . . . . . . . . . 11 ⊢ (𝐶 = +∞ → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
33	32	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
34	31, 33	mpbird 257	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
35	34	adantlr 715	. . . . . . . 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 13150	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
38	37	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < -∞)
39		breq2 5128	. . . . . . . . . . . 12 ⊢ (𝐶 = -∞ → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
40	39	adantl 481	. . . . . . . . . . 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 490	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
44	43	adantll 714	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
45	29, 36, 44	3jaodan 1433	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
46	45	anasss 466	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
47		pnfnlt 13149	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
48	47	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
49		breq1 5127	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
50	49	adantr 480	. . . . . . . . 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 491	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
54	53	adantrr 717	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
55		mnflt 13144	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ → -∞ < 𝐶)
56	55	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → -∞ < 𝐶)
57		breq1 5127	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
58	57	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
59	56, 58	mpbird 257	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → 𝐴 < 𝐶)
60	59	a1d 25	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
61	60	adantlr 715	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
62		mnfltpnf 13147	. . . . . . . . . 10 ⊢ -∞ < +∞
63		breq12 5129	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ -∞ < +∞))
64	62, 63	mpbiri 258	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
65	64	a1d 25	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
66	65	adantlr 715	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
67	43	adantll 714	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
68	61, 66, 67	3jaodan 1433	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
69	68	anasss 466	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
70	46, 54, 69	3jaoian 1432	. . . 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 1407	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
73	1, 72	syl3an1b 1405	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 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5124 ℝcr 11133 +∞cpnf 11271 -∞cmnf 11272 ℝ^*cxr 11273 < clt 11274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-pre-lttrn 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279

This theorem is referenced by: xrltso 13162 xrlelttr 13177 xrltletr 13178 xrlttrd 13180 xrub 13333 ioo0 13392 ioojoin 13505 hashgt23el 14447 leordtval2 23155 icopnfcld 24711 iocmnfcld 24712 ismbf3d 25612 tanord1 26503 tan2h 37641 asindmre 37732 iccpartlt 47405

W3C validator