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Mirrors > Home > MPE Home > Th. List > lerec2d | Structured version Visualization version GIF version |
Description: Reciprocal swap in a 'less than or equal to' relation. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
rpaddcld.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
lerec2d.2 | ⊢ (𝜑 → 𝐴 ≤ (1 / 𝐵)) |
Ref | Expression |
---|---|
lerec2d | ⊢ (𝜑 → 𝐵 ≤ (1 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lerec2d.2 | . 2 ⊢ (𝜑 → 𝐴 ≤ (1 / 𝐵)) | |
2 | rpred.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
3 | 2 | rpregt0d 12162 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
4 | rpaddcld.1 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
5 | 4 | rpregt0d 12162 | . . 3 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
6 | lerec2 11241 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ (1 / 𝐵) ↔ 𝐵 ≤ (1 / 𝐴))) | |
7 | 3, 5, 6 | syl2anc 581 | . 2 ⊢ (𝜑 → (𝐴 ≤ (1 / 𝐵) ↔ 𝐵 ≤ (1 / 𝐴))) |
8 | 1, 7 | mpbid 224 | 1 ⊢ (𝜑 → 𝐵 ≤ (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 0cc0 10252 1c1 10253 < clt 10391 ≤ cle 10392 / cdiv 11009 ℝ+crp 12112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-rp 12113 |
This theorem is referenced by: expcnv 14970 |
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