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Theorem mulgt0con1dlem 39569
Description: Lemma for mulgt0con1d 39570. 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 10637 . . 3 (𝜑 → 0 ∈ ℝ)
31, 2lttrid 10771 . 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 962 . . . 4 (𝜑 → ((𝐴 = 0 ∨ 0 < 𝐴) → (𝐵 = 0 ∨ 0 < 𝐵)))
76con3d 155 . . 3 (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → ¬ (𝐴 = 0 ∨ 0 < 𝐴)))
8 mulgt0con1dlem.a . . . 4 (𝜑𝐴 ∈ ℝ)
98, 2lttrid 10771 . . 3 (𝜑 → (𝐴 < 0 ↔ ¬ (𝐴 = 0 ∨ 0 < 𝐴)))
107, 9sylibrd 262 . 2 (𝜑 → (¬ (𝐵 = 0 ∨ 0 < 𝐵) → 𝐴 < 0))
113, 10sylbid 243 1 (𝜑 → (𝐵 < 0 → 𝐴 < 0))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 844   = wceq 1538  wcel 2112   class class class wbr 5033  cr 10529  0cc0 10530   < clt 10668
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-resscn 10587  ax-1cn 10588  ax-addrcl 10591  ax-rnegex 10601  ax-cnre 10603  ax-pre-lttri 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-ltxr 10673
This theorem is referenced by:  mulgt0con1d  39570  mulgt0con2d  39571
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