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Mirrors > Home > MPE Home > Th. List > lt2sqd | Structured version Visualization version GIF version |
Description: The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
sqgt0d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt2sqd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lt2sqd.3 | ⊢ (𝜑 → 0 ≤ 𝐴) |
lt2sqd.4 | ⊢ (𝜑 → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
lt2sqd | ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqgt0d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt2sqd.3 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
3 | lt2sqd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | lt2sqd.4 | . 2 ⊢ (𝜑 → 0 ≤ 𝐵) | |
5 | lt2sq 14095 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) | |
6 | 1, 2, 3, 4, 5 | syl22anc 838 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 ℝcr 11106 0cc0 11107 < clt 11245 ≤ cle 11246 2c2 12264 ↑cexp 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-n0 12470 df-z 12556 df-uz 12820 df-seq 13964 df-exp 14025 |
This theorem is referenced by: nonsq 16692 pythagtriplem10 16750 pockthg 16836 minveclem3b 24937 minveclem3 24938 minveclem4 24941 tangtx 26007 tanregt0 26040 bndatandm 26424 basellem8 26582 chpub 26713 bposlem6 26782 bposlem7 26783 2sqblem 26924 minvecolem4 30121 minvecolem5 30122 sqsscirc1 32877 areacirclem1 36565 areacirclem4 36568 areacirclem5 36569 areacirc 36570 cntotbnd 36653 rrndstprj2 36688 pell14qrgt0 41583 pell14qrgapw 41600 rmspecnonsq 41631 rmspecfund 41633 rmspecpos 41641 |
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