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Theorem xrlttr 8446
Description: Ordering on the extended reals is transitive. (Contributed by NM, 15-Oct-2005.)
Assertion
Ref Expression
xrlttr ((A * B * 𝐶 *) → ((A < B B < 𝐶) → A < 𝐶))

Proof of Theorem xrlttr
StepHypRef Expression
1 elxr 8426 . 2 (A * ↔ (A A = +∞ A = -∞))
2 elxr 8426 . . 3 (𝐶 * ↔ (𝐶 𝐶 = +∞ 𝐶 = -∞))
3 elxr 8426 . . . . . . . . 9 (B * ↔ (B B = +∞ B = -∞))
4 lttr 6849 . . . . . . . . . . . 12 ((A B 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
543expa 1103 . . . . . . . . . . 11 (((A B ℝ) 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
65an32s 502 . . . . . . . . . 10 (((A 𝐶 ℝ) B ℝ) → ((A < B B < 𝐶) → A < 𝐶))
7 rexr 6828 . . . . . . . . . . . . . . . 16 (𝐶 ℝ → 𝐶 *)
8 pnfnlt 8438 . . . . . . . . . . . . . . . 16 (𝐶 * → ¬ +∞ < 𝐶)
97, 8syl 14 . . . . . . . . . . . . . . 15 (𝐶 ℝ → ¬ +∞ < 𝐶)
109adantr 261 . . . . . . . . . . . . . 14 ((𝐶 B = +∞) → ¬ +∞ < 𝐶)
11 breq1 3758 . . . . . . . . . . . . . . 15 (B = +∞ → (B < 𝐶 ↔ +∞ < 𝐶))
1211adantl 262 . . . . . . . . . . . . . 14 ((𝐶 B = +∞) → (B < 𝐶 ↔ +∞ < 𝐶))
1310, 12mtbird 597 . . . . . . . . . . . . 13 ((𝐶 B = +∞) → ¬ B < 𝐶)
1413pm2.21d 549 . . . . . . . . . . . 12 ((𝐶 B = +∞) → (B < 𝐶A < 𝐶))
1514adantll 445 . . . . . . . . . . 11 (((A 𝐶 ℝ) B = +∞) → (B < 𝐶A < 𝐶))
1615adantld 263 . . . . . . . . . 10 (((A 𝐶 ℝ) B = +∞) → ((A < B B < 𝐶) → A < 𝐶))
17 rexr 6828 . . . . . . . . . . . . . . . 16 (A ℝ → A *)
18 nltmnf 8439 . . . . . . . . . . . . . . . 16 (A * → ¬ A < -∞)
1917, 18syl 14 . . . . . . . . . . . . . . 15 (A ℝ → ¬ A < -∞)
2019adantr 261 . . . . . . . . . . . . . 14 ((A B = -∞) → ¬ A < -∞)
21 breq2 3759 . . . . . . . . . . . . . . 15 (B = -∞ → (A < BA < -∞))
2221adantl 262 . . . . . . . . . . . . . 14 ((A B = -∞) → (A < BA < -∞))
2320, 22mtbird 597 . . . . . . . . . . . . 13 ((A B = -∞) → ¬ A < B)
2423pm2.21d 549 . . . . . . . . . . . 12 ((A B = -∞) → (A < BA < 𝐶))
2524adantlr 446 . . . . . . . . . . 11 (((A 𝐶 ℝ) B = -∞) → (A < BA < 𝐶))
2625adantrd 264 . . . . . . . . . 10 (((A 𝐶 ℝ) B = -∞) → ((A < B B < 𝐶) → A < 𝐶))
276, 16, 263jaodan 1200 . . . . . . . . 9 (((A 𝐶 ℝ) (B B = +∞ B = -∞)) → ((A < B B < 𝐶) → A < 𝐶))
283, 27sylan2b 271 . . . . . . . 8 (((A 𝐶 ℝ) B *) → ((A < B B < 𝐶) → A < 𝐶))
2928an32s 502 . . . . . . 7 (((A B *) 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
30 ltpnf 8432 . . . . . . . . . . 11 (A ℝ → A < +∞)
3130adantr 261 . . . . . . . . . 10 ((A 𝐶 = +∞) → A < +∞)
32 breq2 3759 . . . . . . . . . . 11 (𝐶 = +∞ → (A < 𝐶A < +∞))
3332adantl 262 . . . . . . . . . 10 ((A 𝐶 = +∞) → (A < 𝐶A < +∞))
3431, 33mpbird 156 . . . . . . . . 9 ((A 𝐶 = +∞) → A < 𝐶)
3534adantlr 446 . . . . . . . 8 (((A B *) 𝐶 = +∞) → A < 𝐶)
3635a1d 22 . . . . . . 7 (((A B *) 𝐶 = +∞) → ((A < B B < 𝐶) → A < 𝐶))
37 nltmnf 8439 . . . . . . . . . . . 12 (B * → ¬ B < -∞)
3837adantr 261 . . . . . . . . . . 11 ((B * 𝐶 = -∞) → ¬ B < -∞)
39 breq2 3759 . . . . . . . . . . . 12 (𝐶 = -∞ → (B < 𝐶B < -∞))
4039adantl 262 . . . . . . . . . . 11 ((B * 𝐶 = -∞) → (B < 𝐶B < -∞))
4138, 40mtbird 597 . . . . . . . . . 10 ((B * 𝐶 = -∞) → ¬ B < 𝐶)
4241pm2.21d 549 . . . . . . . . 9 ((B * 𝐶 = -∞) → (B < 𝐶A < 𝐶))
4342adantld 263 . . . . . . . 8 ((B * 𝐶 = -∞) → ((A < B B < 𝐶) → A < 𝐶))
4443adantll 445 . . . . . . 7 (((A B *) 𝐶 = -∞) → ((A < B B < 𝐶) → A < 𝐶))
4529, 36, 443jaodan 1200 . . . . . 6 (((A B *) (𝐶 𝐶 = +∞ 𝐶 = -∞)) → ((A < B B < 𝐶) → A < 𝐶))
4645anasss 379 . . . . 5 ((A (B * (𝐶 𝐶 = +∞ 𝐶 = -∞))) → ((A < B B < 𝐶) → A < 𝐶))
47 pnfnlt 8438 . . . . . . . . . 10 (B * → ¬ +∞ < B)
4847adantl 262 . . . . . . . . 9 ((A = +∞ B *) → ¬ +∞ < B)
49 breq1 3758 . . . . . . . . . 10 (A = +∞ → (A < B ↔ +∞ < B))
5049adantr 261 . . . . . . . . 9 ((A = +∞ B *) → (A < B ↔ +∞ < B))
5148, 50mtbird 597 . . . . . . . 8 ((A = +∞ B *) → ¬ A < B)
5251pm2.21d 549 . . . . . . 7 ((A = +∞ B *) → (A < BA < 𝐶))
5352adantrd 264 . . . . . 6 ((A = +∞ B *) → ((A < B B < 𝐶) → A < 𝐶))
5453adantrr 448 . . . . 5 ((A = +∞ (B * (𝐶 𝐶 = +∞ 𝐶 = -∞))) → ((A < B B < 𝐶) → A < 𝐶))
55 mnflt 8434 . . . . . . . . . . 11 (𝐶 ℝ → -∞ < 𝐶)
5655adantl 262 . . . . . . . . . 10 ((A = -∞ 𝐶 ℝ) → -∞ < 𝐶)
57 breq1 3758 . . . . . . . . . . 11 (A = -∞ → (A < 𝐶 ↔ -∞ < 𝐶))
5857adantr 261 . . . . . . . . . 10 ((A = -∞ 𝐶 ℝ) → (A < 𝐶 ↔ -∞ < 𝐶))
5956, 58mpbird 156 . . . . . . . . 9 ((A = -∞ 𝐶 ℝ) → A < 𝐶)
6059a1d 22 . . . . . . . 8 ((A = -∞ 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
6160adantlr 446 . . . . . . 7 (((A = -∞ B *) 𝐶 ℝ) → ((A < B B < 𝐶) → A < 𝐶))
62 mnfltpnf 8436 . . . . . . . . . 10 -∞ < +∞
63 breq12 3760 . . . . . . . . . 10 ((A = -∞ 𝐶 = +∞) → (A < 𝐶 ↔ -∞ < +∞))
6462, 63mpbiri 157 . . . . . . . . 9 ((A = -∞ 𝐶 = +∞) → A < 𝐶)
6564a1d 22 . . . . . . . 8 ((A = -∞ 𝐶 = +∞) → ((A < B B < 𝐶) → A < 𝐶))
6665adantlr 446 . . . . . . 7 (((A = -∞ B *) 𝐶 = +∞) → ((A < B B < 𝐶) → A < 𝐶))
6743adantll 445 . . . . . . 7 (((A = -∞ B *) 𝐶 = -∞) → ((A < B B < 𝐶) → A < 𝐶))
6861, 66, 673jaodan 1200 . . . . . 6 (((A = -∞ B *) (𝐶 𝐶 = +∞ 𝐶 = -∞)) → ((A < B B < 𝐶) → A < 𝐶))
6968anasss 379 . . . . 5 ((A = -∞ (B * (𝐶 𝐶 = +∞ 𝐶 = -∞))) → ((A < B B < 𝐶) → A < 𝐶))
7046, 54, 693jaoian 1199 . . . 4 (((A A = +∞ A = -∞) (B * (𝐶 𝐶 = +∞ 𝐶 = -∞))) → ((A < B B < 𝐶) → A < 𝐶))
71703impb 1099 . . 3 (((A A = +∞ A = -∞) B * (𝐶 𝐶 = +∞ 𝐶 = -∞)) → ((A < B B < 𝐶) → A < 𝐶))
722, 71syl3an3b 1172 . 2 (((A A = +∞ A = -∞) B * 𝐶 *) → ((A < B B < 𝐶) → A < 𝐶))
731, 72syl3an1b 1170 1 ((A * B * 𝐶 *) → ((A < B B < 𝐶) → A < 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3o 883   w3a 884   = wceq 1242   wcel 1390   class class class wbr 3755  cr 6670  +∞cpnf 6814  -∞cmnf 6815  *cxr 6816   < clt 6817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6734  ax-resscn 6735  ax-pre-lttrn 6757
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6819  df-mnf 6820  df-xr 6821  df-ltxr 6822
This theorem is referenced by:  xrltso  8447  xrlttrd  8455
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