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Mirrors > Home > ILE Home > Th. List > ltmul2 | GIF version |
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) |
Ref | Expression |
---|---|
ltmul2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1 8563 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) | |
2 | recn 7958 | . . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
3 | recn 7958 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | mulcom 7954 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) | |
5 | 3, 4 | sylan 283 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
6 | 5 | 3adant2 1017 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) = (𝐶 · 𝐴)) |
7 | recn 7958 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
8 | mulcom 7954 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) | |
9 | 7, 8 | sylan 283 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
10 | 9 | 3adant1 1016 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
11 | 6, 10 | breq12d 4028 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
12 | 2, 11 | syl3an3 1283 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
13 | 12 | 3adant3r 1236 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) < (𝐵 · 𝐶) ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
14 | 1, 13 | bitrd 188 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐶 · 𝐴) < (𝐶 · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 979 = wceq 1363 ∈ wcel 2158 class class class wbr 4015 (class class class)co 5888 ℂcc 7823 ℝcr 7824 0cc0 7825 · cmul 7830 < clt 8006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-sub 8144 df-neg 8145 |
This theorem is referenced by: ltmul12a 8831 mulgt1 8834 ltmulgt11 8835 lt2msq1 8856 ltdiv2 8858 ltmul2i 8894 ltmul2d 9753 ef01bndlem 11778 cos01gt0 11784 sin4lt0 11788 pockthg 12369 tangtx 14612 lgsdilem 14781 |
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