Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mulgt0con1dlem Structured version   Visualization version   GIF version

Theorem mulgt0con1dlem 41110
Description: Lemma for mulgt0con1d 41111. Contraposes a positive deduction to a negative deduction. (Contributed by SN, 26-Jun-2024.)
Hypotheses
Ref Expression
mulgt0con1dlem.a (𝜑𝐴 ∈ ℝ)
mulgt0con1dlem.b (𝜑𝐵 ∈ ℝ)
mulgt0con1dlem.1 (𝜑 → (0 < 𝐴 → 0 < 𝐵))
mulgt0con1dlem.2 (𝜑 → (𝐴 = 0 → 𝐵 = 0))
Assertion
Ref Expression
mulgt0con1dlem (𝜑 → (𝐵 < 0 → 𝐴 < 0))

Proof of Theorem mulgt0con1dlem
StepHypRef Expression
1 mulgt0con1dlem.b . . 3 (𝜑𝐵 ∈ ℝ)
2 0red 11199 . . 3 (𝜑 → 0 ∈ ℝ)
31, 2lttrid 11334 . 2 (𝜑 → (𝐵 < 0 ↔ ¬ (𝐵 = 0 ∨ 0 < 𝐵)))
4 mulgt0con1dlem.2 . . . . 5 (𝜑 → (𝐴 = 0 → 𝐵 = 0))
5 mulgt0con1dlem.1 . . . . 5 (𝜑 → (0 < 𝐴 → 0 < 𝐵))
64, 5orim12d 963 . . . 4 (𝜑 → ((𝐴 = 0 ∨ 0 < 𝐴) → (𝐵 = 0 ∨ 0 < 𝐵)))
76con3d 152 . . 3 (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → ¬ (𝐴 = 0 ∨ 0 < 𝐴)))
8 mulgt0con1dlem.a . . . 4 (𝜑𝐴 ∈ ℝ)
98, 2lttrid 11334 . . 3 (𝜑 → (𝐴 < 0 ↔ ¬ (𝐴 = 0 ∨ 0 < 𝐴)))
107, 9sylibrd 258 . 2 (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → 𝐴 < 0))
113, 10sylbid 239 1 (𝜑 → (𝐵 < 0 → 𝐴 < 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 845   = wceq 1541  wcel 2106   class class class wbr 5141  cr 11091  0cc0 11092   < clt 11230
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 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-resscn 11149  ax-1cn 11150  ax-addrcl 11153  ax-rnegex 11163  ax-cnre 11165  ax-pre-lttri 11166
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-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-ltxr 11235
This theorem is referenced by:  mulgt0con1d  41111  mulgt0con2d  41112
  Copyright terms: Public domain W3C validator