xrlttr - Metamath Proof Explorer

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Theorem xrlttr 13059

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 13037	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13037	. . 3 ⊢ (𝐶 ∈ ℝ^* ↔ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))
3		elxr 13037	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
4		lttr 11231	. . . . . . . . . . . 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 650	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
7		rexr 11201	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
8		pnfnlt 13049	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ ℝ^* → ¬ +∞ < 𝐶)
9	7, 8	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ ℝ → ¬ +∞ < 𝐶)
10	9	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ +∞ < 𝐶)
11		breq1 5108	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = +∞ → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
12	11	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐶 ↔ +∞ < 𝐶))
13	10, 12	mtbird 324	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → ¬ 𝐵 < 𝐶)
14	13	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 < 𝐶 → 𝐴 < 𝐶))
15	14	adantll 712	. . . . . . . . . . 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 11201	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
18		nltmnf 13050	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
19	17, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
20	19	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
21		breq2 5109	. . . . . . . . . . . . . . 15 ⊢ (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
22	21	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
23	20, 22	mtbird 324	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
24	23	pm2.21d 121	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → 𝐴 < 𝐶))
25	24	adantlr 713	. . . . . . . . . . 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 1430	. . . . . . . . 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 650	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
30		ltpnf 13041	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
31	30	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 = +∞) → 𝐴 < +∞)
32		breq2 5109	. . . . . . . . . . 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 713	. . . . . . . 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 13050	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < -∞)
38	37	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → ¬ 𝐵 < -∞)
39		breq2 5109	. . . . . . . . . . . 12 ⊢ (𝐶 = -∞ → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
40	39	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 = -∞) → (𝐵 < 𝐶 ↔ 𝐵 < -∞))
41	38, 40	mtbird 324	. . . . . . . . . 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 712	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
45	29, 36, 44	3jaodan 1430	. . . . . 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 13049	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
48	47	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
49		breq1 5108	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
50	49	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
51	48, 50	mtbird 324	. . . . . . . 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 715	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
55		mnflt 13044	. . . . . . . . . . 11 ⊢ (𝐶 ∈ ℝ → -∞ < 𝐶)
56	55	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐶 ∈ ℝ) → -∞ < 𝐶)
57		breq1 5108	. . . . . . . . . . 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 713	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
62		mnfltpnf 13047	. . . . . . . . . 10 ⊢ -∞ < +∞
63		breq12 5110	. . . . . . . . . 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 713	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = +∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
67	43	adantll 712	. . . . . . 7 ⊢ (((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 = -∞) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
68	61, 66, 67	3jaodan 1430	. . . . . 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 1429	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞))) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
71	70	3impb 1115	. . 3 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ (𝐶 ∈ ℝ ∨ 𝐶 = +∞ ∨ 𝐶 = -∞)) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
72	2, 71	syl3an3b 1405	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶))
73	1, 72	syl3an1b 1403	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 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ℝcr 11050 +∞cpnf 11186 -∞cmnf 11187 ℝ^*cxr 11188 < clt 11189

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-pre-lttrn 11126

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194

This theorem is referenced by: xrltso 13060 xrlelttr 13075 xrltletr 13076 xrlttrd 13078 xrub 13231 ioo0 13289 ioojoin 13400 hashgt23el 14324 leordtval2 22563 icopnfcld 24131 iocmnfcld 24132 ismbf3d 25018 tanord1 25893 tan2h 36070 asindmre 36161 iccpartlt 45606

W3C validator