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Mirrors > Home > ILE Home > Th. List > ltmulgt12 | GIF version |
Description: Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
Ref | Expression |
---|---|
ltmulgt12 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐵 · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmulgt11 8315 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵))) | |
2 | recn 7465 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
3 | recn 7465 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | mulcom 7461 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | |
5 | 2, 3, 4 | syl2an 283 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
6 | 5 | 3adant3 963 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
7 | 6 | breq2d 3855 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < (𝐴 · 𝐵) ↔ 𝐴 < (𝐵 · 𝐴))) |
8 | 1, 7 | bitrd 186 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐵 · 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 class class class wbr 3843 (class class class)co 5644 ℂcc 7338 ℝcr 7339 0cc0 7340 1c1 7341 · cmul 7345 < clt 7512 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-mulrcl 7434 ax-addcom 7435 ax-mulcom 7436 ax-addass 7437 ax-mulass 7438 ax-distr 7439 ax-i2m1 7440 ax-1rid 7442 ax-0id 7443 ax-rnegex 7444 ax-precex 7445 ax-cnre 7446 ax-pre-ltadd 7451 ax-pre-mulgt0 7452 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-br 3844 df-opab 3898 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-iota 4975 df-fun 5012 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-pnf 7514 df-mnf 7515 df-ltxr 7517 df-sub 7645 df-neg 7646 |
This theorem is referenced by: ltmulgt12d 9200 dvdsnprmd 11372 |
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