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Mirrors > Home > MPE Home > Th. List > lemul1 | Structured version Visualization version GIF version |
Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
Ref | Expression |
---|---|
lemul1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1 12115 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 < 𝐵 ↔ (𝐴 · 𝐶) < (𝐵 · 𝐶))) | |
2 | recn 11243 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | recn 11243 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | recn 11243 | . . . . . . 7 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℂ) | |
5 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → 𝐶 ∈ ℂ) |
6 | gt0ne0 11726 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → 𝐶 ≠ 0) | |
7 | 5, 6 | jca 511 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 0 < 𝐶) → (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) |
8 | mulcan2 11899 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) | |
9 | 2, 3, 7, 8 | syl3an 1159 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵)) |
10 | 9 | bicomd 223 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 = 𝐵 ↔ (𝐴 · 𝐶) = (𝐵 · 𝐶))) |
11 | 1, 10 | orbi12d 918 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵) ↔ ((𝐴 · 𝐶) < (𝐵 · 𝐶) ∨ (𝐴 · 𝐶) = (𝐵 · 𝐶)))) |
12 | leloe 11345 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) | |
13 | 12 | 3adant3 1131 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴 = 𝐵))) |
14 | remulcl 11238 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · 𝐶) ∈ ℝ) | |
15 | 14 | 3adant2 1130 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · 𝐶) ∈ ℝ) |
16 | remulcl 11238 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℝ) | |
17 | 16 | 3adant1 1129 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℝ) |
18 | 15, 17 | leloed 11402 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 · 𝐶) ≤ (𝐵 · 𝐶) ↔ ((𝐴 · 𝐶) < (𝐵 · 𝐶) ∨ (𝐴 · 𝐶) = (𝐵 · 𝐶)))) |
19 | 18 | 3adant3r 1180 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) ≤ (𝐵 · 𝐶) ↔ ((𝐴 · 𝐶) < (𝐵 · 𝐶) ∨ (𝐴 · 𝐶) = (𝐵 · 𝐶)))) |
20 | 11, 13, 19 | 3bitr4d 311 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐶) ≤ (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 · cmul 11158 < clt 11293 ≤ cle 11294 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: lemul2 12118 lemul1a 12119 lediv23 12158 lemul1i 12188 ledivp1i 12191 div4p1lem1div2 12519 lemul1d 13118 xlemul1a 13327 iccdil 13527 expgt1 14138 sqlecan 14245 facubnd 14336 01sqrexlem2 15279 01sqrexlem6 15283 eirrlem 16237 mbfi1fseqlem3 25767 mbfi1fseqlem4 25768 mbfi1fseqlem5 25769 itg2monolem3 25802 atans2 26989 log2tlbnd 27003 fsumfldivdiaglem 27247 chtublem 27270 bposlem2 27344 bposlem5 27347 gausslemma2dlem2 27426 2lgslem1a1 27448 selberglem2 27605 pntpbnd1a 27644 pntpbnd2 27646 ostth2lem3 27694 htthlem 30946 cnlnadjlem7 32102 bfplem1 37809 jm2.24nn 42948 jm3.1lem2 43007 stoweidlem14 45970 stoweidlem26 45982 stoweidlem34 45990 fmtno4prmfac 47497 |
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