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Mirrors > Home > MPE Home > Th. List > recgt1 | Structured version Visualization version GIF version |
Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005.) |
Ref | Expression |
---|---|
recgt1 | ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐴 ↔ (1 / 𝐴) < 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10251 | . . 3 ⊢ 1 ∈ ℝ | |
2 | 0lt1 10762 | . . 3 ⊢ 0 < 1 | |
3 | ltrec 11117 | . . 3 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 < 𝐴 ↔ (1 / 𝐴) < (1 / 1))) | |
4 | 1, 2, 3 | mpanl12 720 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐴 ↔ (1 / 𝐴) < (1 / 1))) |
5 | 1div1e1 10929 | . . 3 ⊢ (1 / 1) = 1 | |
6 | 5 | breq2i 4812 | . 2 ⊢ ((1 / 𝐴) < (1 / 1) ↔ (1 / 𝐴) < 1) |
7 | 4, 6 | syl6bb 276 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐴 ↔ (1 / 𝐴) < 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2139 class class class wbr 4804 (class class class)co 6814 ℝcr 10147 0cc0 10148 1c1 10149 < clt 10286 / cdiv 10896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 |
This theorem is referenced by: recgt1i 11132 recreclt 11134 recgt1d 12099 fldiv4p1lem1div2 12850 0.999... 14831 0.999...OLD 14832 rpnnen2lem3 15164 rpnnen2lem9 15170 flodddiv4 15359 log2cnv 24891 rplogsumlem2 25394 rpvmasumlem 25396 pntibndlem1 25498 |
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