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Theorem mulgt0con1dlem 41632
Description: Lemma for mulgt0con1d 41633. 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 11221 . . 3 (𝜑 → 0 ∈ ℝ)
31, 2lttrid 11356 . 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 961 . . . 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 11356 . . 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 843   = wceq 1539  wcel 2104   class class class wbr 5147  cr 11111  0cc0 11112   < clt 11252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-resscn 11169  ax-1cn 11170  ax-addrcl 11173  ax-rnegex 11183  ax-cnre 11185  ax-pre-lttri 11186
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-ltxr 11257
This theorem is referenced by:  mulgt0con1d  41633  mulgt0con2d  41634
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