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Mirrors > Home > MPE Home > Th. List > lt0neg1d | Structured version Visualization version GIF version |
Description: Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
lt0neg1d | ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt0neg1 11303 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 < 0 ↔ 0 < -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2112 class class class wbr 5039 ℝcr 10693 0cc0 10694 < clt 10832 -cneg 11028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 |
This theorem is referenced by: recgt0 11643 prodge0rd 12658 expneg 13608 discr1 13771 bitsfzo 15957 subgmulg 18511 xrhmeo 23797 mbfmulc2lem 24498 mbfposr 24503 dvferm2lem 24837 dvferm2 24838 sincosq4sgn 25345 tanabsge 25350 sinq34lt0t 25353 tanord 25381 argimlt0 25455 logdmnrp 25483 atanlogsub 25753 atantan 25760 lgsdilem 26159 ostth3 26473 sgnmul 32175 fdvneggt 32246 oexpreposd 39969 elpell14qr2 40328 jm2.24 40429 mulltgt0 42179 lt0neg1dd 42541 fourierdlem43 43309 fourierdlem44 43310 fourierdlem92 43357 fourierdlem109 43374 requad01 44689 |
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