![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ltmulgt11 | GIF version |
Description: Multiplication by a number greater than 1. (Contributed by NM, 24-Dec-2005.) |
Ref | Expression |
---|---|
ltmulgt11 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7986 | . . . . 5 ⊢ 1 ∈ ℝ | |
2 | ltmul2 8843 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 < 𝐵 ↔ (𝐴 · 1) < (𝐴 · 𝐵))) | |
3 | 1, 2 | mp3an1 1335 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 < 𝐵 ↔ (𝐴 · 1) < (𝐴 · 𝐵))) |
4 | 3 | 3impb 1201 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ (𝐴 · 1) < (𝐴 · 𝐵))) |
5 | 4 | 3com12 1209 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ (𝐴 · 1) < (𝐴 · 𝐵))) |
6 | ax-1rid 7948 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
7 | 6 | 3ad2ant1 1020 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 · 1) = 𝐴) |
8 | 7 | breq1d 4028 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → ((𝐴 · 1) < (𝐴 · 𝐵) ↔ 𝐴 < (𝐴 · 𝐵))) |
9 | 5, 8 | bitrd 188 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 < 𝐴) → (1 < 𝐵 ↔ 𝐴 < (𝐴 · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5896 ℝcr 7840 0cc0 7841 1c1 7842 · cmul 7846 < clt 8022 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-ltxr 8027 df-sub 8160 df-neg 8161 |
This theorem is referenced by: ltmulgt12 8852 ltmulgt11d 9762 efltim 11738 nprm 12155 |
Copyright terms: Public domain | W3C validator |