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Mirrors > Home > MPE Home > Th. List > ltrecii | Structured version Visualization version GIF version |
Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.) |
Ref | Expression |
---|---|
ltplus1.1 | ⊢ 𝐴 ∈ ℝ |
prodgt0.2 | ⊢ 𝐵 ∈ ℝ |
ltreci.3 | ⊢ 0 < 𝐴 |
ltreci.4 | ⊢ 0 < 𝐵 |
Ref | Expression |
---|---|
ltrecii | ⊢ (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltreci.3 | . 2 ⊢ 0 < 𝐴 | |
2 | ltreci.4 | . 2 ⊢ 0 < 𝐵 | |
3 | ltplus1.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
4 | prodgt0.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
5 | 3, 4 | ltreci 11549 | . 2 ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴))) |
6 | 1, 2, 5 | mp2an 690 | 1 ⊢ (𝐴 < 𝐵 ↔ (1 / 𝐵) < (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 ℝcr 10535 0cc0 10536 1c1 10537 < clt 10674 / cdiv 11296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 |
This theorem is referenced by: pcoass 23627 chebbnd1 26047 subfacval3 32436 stoweid 42347 |
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