xrlttr - Metamath Proof Explorer

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Theorem xrlttr 13166

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 13142	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13142	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
3		elxr 13142	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4		lttr 11287	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
5	4	3expa 1136	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
6	5	an32s 664	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7		rexr 11256	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
8		pnfnlt 13154	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ^* → ¬ +∞ < 𝐶)
9	7, 8	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℝ → ¬ +∞ < 𝐶)
10	9	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐶)
11		breq1 5113	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = +∞ → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
12	11	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
13	10, 12	mtbird 328	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐶)
14	13	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
15	14	adantll 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
16	15	adantld 495	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
17		rexr 11256	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
18		nltmnf 13155	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
19	17, 18	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
20	19	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
21		breq2 5114	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
22	21	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
23	20, 22	mtbird 328	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
24	23	pm2.21d 122	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
25	24	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
26	25	adantrd 496	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
27	6, 16, 26	3jaodan 1458	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
28	3, 27	sylan2b 605	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
29	28	an32s 664	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
30		ltpnf 13146	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
31	30	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < +∞)
32		breq2 5114	. . . . . . . . . . 11 ⊢ (𝐶 = +∞ → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
33	32	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
34	31, 33	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
35	34	adantlr 727	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
36	35	a1d 26	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
37		nltmnf 13155	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
38	37	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < -∞)
39		breq2 5114	. . . . . . . . . . . 12 ⊢ (𝐶 = -∞ → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
40	39	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
41	38, 40	mtbird 328	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < 𝐶)
42	41	pm2.21d 122	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
43	42	adantld 495	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
44	43	adantll 726	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
45	29, 36, 44	3jaodan 1458	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
46	45	anasss 471	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
47		pnfnlt 13154	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
48	47	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
49		breq1 5113	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
50	49	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
51	48, 50	mtbird 328	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ 𝐴 < 𝐵)
52	51	pm2.21d 122	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
53	52	adantrd 496	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
54	53	adantrr 729	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
55		mnflt 13149	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ → -∞ < 𝐶)
56	55	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → -∞ < 𝐶)
57		breq1 5113	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
58	57	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
59	56, 58	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → 𝐴 < 𝐶)
60	59	a1d 26	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
61	60	adantlr 727	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
62		mnfltpnf 13152	. . . . . . . . . 10 ⊢ -∞ < +∞
63		breq12 5115	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ -∞ < +∞))
64	62, 63	mpbiri 261	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
65	64	a1d 26	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
66	65	adantlr 727	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
67	43	adantll 726	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
68	61, 66, 67	3jaodan 1458	. . . . . 6 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
69	68	anasss 471	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
70	46, 54, 69	3jaoian 1457	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
71	70	3impb 1132	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
72	2, 71	syl3an3b 1432	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
73	1, 72	syl3an1b 1430	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 209 ∧ wa 400 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 class class class wbr 5110 ℝcr 11100 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-pre-lttrn 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249

This theorem is referenced by: xrltso 13167 xrlelttr 13182 xrltletr 13183 xrlttrd 13185 xrub 13339 ioo0 13398 ioojoin 13511 hashgt23el 14463 leordtval2 23350 icopnfcld 24905 iocmnfcld 24906 ismbf3d 25794 tanord1 26683 tan2h 38244 asindmre 38335 iccpartlt 48156

W3C validator