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Mirrors > Home > MPE Home > Th. List > mulle0b | Structured version Visualization version GIF version |
Description: A condition for multiplication to be nonpositive. (Contributed by Scott Fenton, 25-Jun-2013.) |
Ref | Expression |
---|---|
mulle0b | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴 ∧ 𝐵 ≤ 0)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | remulcl 10021 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) | |
2 | 1 | le0neg1d 10599 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) ≤ 0 ↔ 0 ≤ -(𝐴 · 𝐵))) |
3 | le0neg2 10537 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (0 ≤ 𝐵 ↔ -𝐵 ≤ 0)) | |
4 | 3 | anbi2d 740 | . . . . 5 ⊢ (𝐵 ∈ ℝ → ((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ↔ (𝐴 ≤ 0 ∧ -𝐵 ≤ 0))) |
5 | le0neg1 10536 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵)) | |
6 | 5 | anbi2d 740 | . . . . 5 ⊢ (𝐵 ∈ ℝ → ((0 ≤ 𝐴 ∧ 𝐵 ≤ 0) ↔ (0 ≤ 𝐴 ∧ 0 ≤ -𝐵))) |
7 | 4, 6 | orbi12d 746 | . . . 4 ⊢ (𝐵 ∈ ℝ → (((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴 ∧ 𝐵 ≤ 0)) ↔ ((𝐴 ≤ 0 ∧ -𝐵 ≤ 0) ∨ (0 ≤ 𝐴 ∧ 0 ≤ -𝐵)))) |
8 | 7 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴 ∧ 𝐵 ≤ 0)) ↔ ((𝐴 ≤ 0 ∧ -𝐵 ≤ 0) ∨ (0 ≤ 𝐴 ∧ 0 ≤ -𝐵)))) |
9 | renegcl 10344 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
10 | mulge0b 10893 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (0 ≤ (𝐴 · -𝐵) ↔ ((𝐴 ≤ 0 ∧ -𝐵 ≤ 0) ∨ (0 ≤ 𝐴 ∧ 0 ≤ -𝐵)))) | |
11 | 9, 10 | sylan2 491 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 · -𝐵) ↔ ((𝐴 ≤ 0 ∧ -𝐵 ≤ 0) ∨ (0 ≤ 𝐴 ∧ 0 ≤ -𝐵)))) |
12 | recn 10026 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
13 | recn 10026 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
14 | mulneg2 10467 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · -𝐵) = -(𝐴 · 𝐵)) | |
15 | 14 | breq2d 4665 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 ≤ (𝐴 · -𝐵) ↔ 0 ≤ -(𝐴 · 𝐵))) |
16 | 12, 13, 15 | syl2an 494 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 · -𝐵) ↔ 0 ≤ -(𝐴 · 𝐵))) |
17 | 8, 11, 16 | 3bitr2rd 297 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ -(𝐴 · 𝐵) ↔ ((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴 ∧ 𝐵 ≤ 0)))) |
18 | 2, 17 | bitrd 268 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 · 𝐵) ≤ 0 ↔ ((𝐴 ≤ 0 ∧ 0 ≤ 𝐵) ∨ (0 ≤ 𝐴 ∧ 𝐵 ≤ 0)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 ≤ cle 10075 -cneg 10267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 |
This theorem is referenced by: mulsuble0b 10895 addmodlteq 12745 colinearalglem4 25789 reclt0d 39607 |
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