Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > lemulge12d | GIF version |
Description: Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
divgt0d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
lemulge11d.3 | ⊢ (𝜑 → 0 ≤ 𝐴) |
lemulge11d.4 | ⊢ (𝜑 → 1 ≤ 𝐵) |
Ref | Expression |
---|---|
lemulge12d | ⊢ (𝜑 → 𝐴 ≤ (𝐵 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltp1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | divgt0d.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | lemulge11d.3 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
4 | lemulge11d.4 | . 2 ⊢ (𝜑 → 1 ≤ 𝐵) | |
5 | lemulge12 8632 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 1 ≤ 𝐵)) → 𝐴 ≤ (𝐵 · 𝐴)) | |
6 | 1, 2, 3, 4, 5 | syl22anc 1217 | 1 ⊢ (𝜑 → 𝐴 ≤ (𝐵 · 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7626 0cc0 7627 1c1 7628 · cmul 7632 ≤ cle 7808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 |
This theorem is referenced by: bernneq3 10421 eflegeo 11415 dvdslelemd 11548 oddennn 11912 |
Copyright terms: Public domain | W3C validator |