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Mirrors > Home > ILE Home > Th. List > lt2mul2divd | GIF version |
Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
lt2mul2divd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
lt2mul2divd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
lt2mul2divd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2mul2divd.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
Ref | Expression |
---|---|
lt2mul2divd | ⊢ (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2mul2divd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lt2mul2divd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | 2 | rpregt0d 9075 | . 2 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 0 < 𝐵)) |
4 | lt2mul2divd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | lt2mul2divd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ+) | |
6 | 5 | rpregt0d 9075 | . 2 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 < 𝐷)) |
7 | lt2mul2div 8234 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) ∧ (𝐶 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷))) → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) | |
8 | 1, 3, 4, 6, 7 | syl22anc 1171 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) < (𝐶 · 𝐷) ↔ (𝐴 / 𝐷) < (𝐶 / 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5591 ℝcr 7252 0cc0 7253 · cmul 7258 < clt 7425 / cdiv 8037 ℝ+crp 9029 |
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 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 |
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-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4084 df-po 4087 df-iso 4088 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-iota 4934 df-fun 4971 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-rp 9030 |
This theorem is referenced by: qtri3or 9543 cvg1nlemcxze 10242 cvg1nlemcau 10244 |
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