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Mirrors > Home > ILE Home > Th. List > lt2msqi | GIF version |
Description: The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.) |
Ref | Expression |
---|---|
ltplus1.1 | ⊢ 𝐴 ∈ ℝ |
prodgt0.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
lt2msqi | ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltplus1.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | prodgt0.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
3 | lt2msq 8101 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵))) | |
4 | 2, 3 | mpanr1 428 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵))) |
5 | 1, 4 | mpanl1 425 | 1 ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐴) < (𝐵 · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1434 class class class wbr 3805 (class class class)co 5564 ℝcr 7112 0cc0 7113 · cmul 7118 < clt 7285 ≤ cle 7286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7199 ax-resscn 7200 ax-1cn 7201 ax-1re 7202 ax-icn 7203 ax-addcl 7204 ax-addrcl 7205 ax-mulcl 7206 ax-mulrcl 7207 ax-addcom 7208 ax-mulcom 7209 ax-addass 7210 ax-mulass 7211 ax-distr 7212 ax-i2m1 7213 ax-0lt1 7214 ax-1rid 7215 ax-0id 7216 ax-rnegex 7217 ax-precex 7218 ax-cnre 7219 ax-pre-ltirr 7220 ax-pre-ltwlin 7221 ax-pre-lttrn 7222 ax-pre-apti 7223 ax-pre-ltadd 7224 ax-pre-mulgt0 7225 ax-pre-mulext 7226 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-br 3806 df-opab 3860 df-id 4076 df-po 4079 df-iso 4080 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-iota 4917 df-fun 4954 df-fv 4960 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-pnf 7287 df-mnf 7288 df-xr 7289 df-ltxr 7290 df-le 7291 df-sub 7418 df-neg 7419 df-reap 7812 df-ap 7819 |
This theorem is referenced by: lt2sqi 9730 |
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