xrlttr - Metamath Proof Explorer

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Theorem xrlttr 13176

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 13152	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13152	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
3		elxr 13152	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4		lttr 11297	. . . . . . . . . . . 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 665	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7		rexr 11266	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
8		pnfnlt 13164	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ^* → ¬ +∞ < 𝐶)
9	7, 8	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℝ → ¬ +∞ < 𝐶)
10	9	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐶)
11		breq1 5114	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = +∞ → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
12	11	adantl 487	. . . . . . . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
16	15	adantld 496	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
17		rexr 11266	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
18		nltmnf 13165	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
19	17, 18	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
20	19	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
21		breq2 5115	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
22	21	adantl 487	. . . . . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
26	25	adantrd 497	. . . . . . . . . 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 606	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
29	28	an32s 665	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
30		ltpnf 13156	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
31	30	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < +∞)
32		breq2 5115	. . . . . . . . . . 11 ⊢ (𝐶 = +∞ → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
33	32	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → (𝐴 < 𝐶 ↔ 𝐴 < +∞))
34	31, 33	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < 𝐶)
35	34	adantlr 728	. . . . . . . 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 13165	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
38	37	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < -∞)
39		breq2 5115	. . . . . . . . . . . 12 ⊢ (𝐶 = -∞ → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
40	39	adantl 487	. . . . . . . . . . 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 496	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
44	43	adantll 727	. . . . . . 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 472	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
47		pnfnlt 13164	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
48	47	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
49		breq1 5114	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
50	49	adantr 486	. . . . . . . . 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 497	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
54	53	adantrr 730	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
55		mnflt 13159	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ → -∞ < 𝐶)
56	55	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → -∞ < 𝐶)
57		breq1 5114	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐴 < 𝐶 ↔ -∞ < 𝐶))
58	57	adantr 486	. . . . . . . . . 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 728	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
62		mnfltpnf 13162	. . . . . . . . . 10 ⊢ -∞ < +∞
63		breq12 5116	. . . . . . . . . 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 728	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
67	43	adantll 727	. . . . . . 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 472	. . . . 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 class class class wbr 5111 ℝcr 11110 +∞cpnf 11251 -∞cmnf 11252 ℝ^*cxr 11253 < clt 11254

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-pre-lttrn 11186

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 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 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259

This theorem is used by: xrltso 13177 xrlelttr 13192 xrltletr 13193 xrlttrd 13195 xrub 13349 ioo0 13408 ioojoin 13521 hashgt23el 14474 leordtval2 23398 icopnfcld 24953 iocmnfcld 24954 ismbf3d 25842 tanord1 26731 tan2h 38296 asindmre 38387 iccpartlt 48206

W3C validator