MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltord1 Structured version   Visualization version   GIF version

Theorem ltord1 10406
Description: Infer an ordering relation from a proof in only one direction. (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
ltord1 ((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 ltord1
StepHypRef Expression
1 ltord.1 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
2 ltord.2 . . 3 (𝑥 = 𝐶𝐴 = 𝑀)
3 ltord.3 . . 3 (𝑥 = 𝐷𝐴 = 𝑁)
4 ltord.4 . . 3 𝑆 ⊆ ℝ
5 ltord.5 . . 3 ((𝜑𝑥𝑆) → 𝐴 ∈ ℝ)
6 ltord.6 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 < 𝑦𝐴 < 𝐵))
71, 2, 3, 4, 5, 6ltordlem 10405 . 2 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
8 eqeq1 2614 . . . . . . . 8 (𝑥 = 𝐶 → (𝑥 = 𝐷𝐶 = 𝐷))
92eqeq1d 2612 . . . . . . . 8 (𝑥 = 𝐶 → (𝐴 = 𝑁𝑀 = 𝑁))
108, 9imbi12d 333 . . . . . . 7 (𝑥 = 𝐶 → ((𝑥 = 𝐷𝐴 = 𝑁) ↔ (𝐶 = 𝐷𝑀 = 𝑁)))
1110, 3vtoclg 3239 . . . . . 6 (𝐶𝑆 → (𝐶 = 𝐷𝑀 = 𝑁))
1211ad2antrl 760 . . . . 5 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 = 𝐷𝑀 = 𝑁))
131, 3, 2, 4, 5, 6ltordlem 10405 . . . . . 6 ((𝜑 ∧ (𝐷𝑆𝐶𝑆)) → (𝐷 < 𝐶𝑁 < 𝑀))
1413ancom2s 840 . . . . 5 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐷 < 𝐶𝑁 < 𝑀))
1512, 14orim12d 879 . . . 4 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → ((𝐶 = 𝐷𝐷 < 𝐶) → (𝑀 = 𝑁𝑁 < 𝑀)))
1615con3d 147 . . 3 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (¬ (𝑀 = 𝑁𝑁 < 𝑀) → ¬ (𝐶 = 𝐷𝐷 < 𝐶)))
175ralrimiva 2949 . . . . . 6 (𝜑 → ∀𝑥𝑆 𝐴 ∈ ℝ)
182eleq1d 2672 . . . . . . 7 (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ))
1918rspccva 3281 . . . . . 6 ((∀𝑥𝑆 𝐴 ∈ ℝ ∧ 𝐶𝑆) → 𝑀 ∈ ℝ)
2017, 19sylan 487 . . . . 5 ((𝜑𝐶𝑆) → 𝑀 ∈ ℝ)
213eleq1d 2672 . . . . . . 7 (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ))
2221rspccva 3281 . . . . . 6 ((∀𝑥𝑆 𝐴 ∈ ℝ ∧ 𝐷𝑆) → 𝑁 ∈ ℝ)
2317, 22sylan 487 . . . . 5 ((𝜑𝐷𝑆) → 𝑁 ∈ ℝ)
2420, 23anim12dan 878 . . . 4 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ))
25 axlttri 9961 . . . 4 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 ↔ ¬ (𝑀 = 𝑁𝑁 < 𝑀)))
2624, 25syl 17 . . 3 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝑀 < 𝑁 ↔ ¬ (𝑀 = 𝑁𝑁 < 𝑀)))
274sseli 3564 . . . . 5 (𝐶𝑆𝐶 ∈ ℝ)
284sseli 3564 . . . . 5 (𝐷𝑆𝐷 ∈ ℝ)
29 axlttri 9961 . . . . 5 ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 < 𝐷 ↔ ¬ (𝐶 = 𝐷𝐷 < 𝐶)))
3027, 28, 29syl2an 493 . . . 4 ((𝐶𝑆𝐷𝑆) → (𝐶 < 𝐷 ↔ ¬ (𝐶 = 𝐷𝐷 < 𝐶)))
3130adantl 481 . . 3 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷 ↔ ¬ (𝐶 = 𝐷𝐷 < 𝐶)))
3216, 26, 313imtr4d 282 . 2 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝑀 < 𝑁𝐶 < 𝐷))
337, 32impbid 201 1 ((𝜑 ∧ (𝐶𝑆𝐷𝑆)) → (𝐶 < 𝐷𝑀 < 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540   class class class wbr 4578  cr 9792   < clt 9931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-resscn 9850  ax-pre-lttri 9867
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-ltxr 9936
This theorem is referenced by:  leord1  10407  ltord2  10409  ltexp2  12734  eflt  14635  tanord1  24032  tanord  24033  monotuz  36318  monotoddzzfi  36319  rpexpmord  36325
  Copyright terms: Public domain W3C validator