MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltordlem Structured version   Visualization version   GIF version

Theorem ltordlem 11676
Description: Lemma for ltord1 11677. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ltord.1 (𝑥 = 𝑦𝐴 = 𝐵)
ltord.2 (𝑥 = 𝐶𝐴 = 𝑀)
ltord.3 (𝑥 = 𝐷𝐴 = 𝑁)
ltord.4 𝑆 ⊆ ℝ
ltord.5 ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)
ltord.6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))
Assertion
Ref Expression
ltordlem ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem ltordlem
StepHypRef Expression
1 ltord.6 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))
21ralrimivva 3195 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵))
3 breq1 5106 . . . 4 (𝑥 = 𝐶 → (𝑥 < 𝑦𝐶 < 𝑦))
4 ltord.2 . . . . 5 (𝑥 = 𝐶𝐴 = 𝑀)
54breq1d 5113 . . . 4 (𝑥 = 𝐶 → (𝐴 < 𝐵𝑀 < 𝐵))
63, 5imbi12d 344 . . 3 (𝑥 = 𝐶 → ((𝑥 < 𝑦𝐴 < 𝐵) ↔ (𝐶 < 𝑦𝑀 < 𝐵)))
7 breq2 5107 . . . 4 (𝑦 = 𝐷 → (𝐶 < 𝑦𝐶 < 𝐷))
8 eqeq1 2740 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = 𝐷𝑦 = 𝐷))
9 ltord.1 . . . . . . . 8 (𝑥 = 𝑦𝐴 = 𝐵)
109eqeq1d 2738 . . . . . . 7 (𝑥 = 𝑦 → (𝐴 = 𝑁𝐵 = 𝑁))
118, 10imbi12d 344 . . . . . 6 (𝑥 = 𝑦 → ((𝑥 = 𝐷𝐴 = 𝑁) ↔ (𝑦 = 𝐷𝐵 = 𝑁)))
12 ltord.3 . . . . . 6 (𝑥 = 𝐷𝐴 = 𝑁)
1311, 12chvarvv 2002 . . . . 5 (𝑦 = 𝐷𝐵 = 𝑁)
1413breq2d 5115 . . . 4 (𝑦 = 𝐷 → (𝑀 < 𝐵𝑀 < 𝑁))
157, 14imbi12d 344 . . 3 (𝑦 = 𝐷 → ((𝐶 < 𝑦𝑀 < 𝐵) ↔ (𝐶 < 𝐷𝑀 < 𝑁)))
166, 15rspc2v 3588 . 2 ((𝐶𝑆𝐷𝑆) → (∀𝑥𝑆𝑦𝑆 (𝑥 < 𝑦𝐴 < 𝐵) → (𝐶 < 𝐷𝑀 < 𝑁)))
172, 16mpan9 507 1 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  wss 3908   class class class wbr 5103  cr 11046   < clt 11185
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-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104
This theorem is referenced by:  ltord1  11677
  Copyright terms: Public domain W3C validator