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Mirrors > Home > ILE Home > Th. List > lemul1 | 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 7376 | . . . 4 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (B < A ↔ (B · 𝐶) < (A · 𝐶))) | |
2 | 1 | notbid 591 | . . 3 ⊢ ((B ∈ ℝ ∧ A ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (¬ B < A ↔ ¬ (B · 𝐶) < (A · 𝐶))) |
3 | 2 | 3com12 1107 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (¬ B < A ↔ ¬ (B · 𝐶) < (A · 𝐶))) |
4 | lenlt 6891 | . . 3 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (A ≤ B ↔ ¬ B < A)) | |
5 | 4 | 3adant3 923 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A ≤ B ↔ ¬ B < A)) |
6 | simp1 903 | . . . 4 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → A ∈ ℝ) | |
7 | simp3l 931 | . . . 4 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ) | |
8 | 6, 7 | remulcld 6853 | . . 3 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A · 𝐶) ∈ ℝ) |
9 | simp2 904 | . . . 4 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → B ∈ ℝ) | |
10 | 9, 7 | remulcld 6853 | . . 3 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (B · 𝐶) ∈ ℝ) |
11 | 8, 10 | lenltd 6931 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → ((A · 𝐶) ≤ (B · 𝐶) ↔ ¬ (B · 𝐶) < (A · 𝐶))) |
12 | 3, 5, 11 | 3bitr4d 209 | 1 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 < 𝐶)) → (A ≤ B ↔ (A · 𝐶) ≤ (B · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℝcr 6710 0cc0 6711 · cmul 6716 < clt 6857 ≤ cle 6858 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulrcl 6782 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-precex 6793 ax-cnre 6794 ax-pre-ltadd 6799 ax-pre-mulgt0 6800 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 |
This theorem is referenced by: lemul2 7604 lediv23 7640 lemul1i 7671 lemul1d 8436 iccdil 8636 expgt1 8947 |
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