Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > leadd1i | GIF version |
Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.) |
Ref | Expression |
---|---|
lt2.1 | ⊢ 𝐴 ∈ ℝ |
lt2.2 | ⊢ 𝐵 ∈ ℝ |
lt2.3 | ⊢ 𝐶 ∈ ℝ |
Ref | Expression |
---|---|
leadd1i | ⊢ (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | lt2.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | lt2.3 | . 2 ⊢ 𝐶 ∈ ℝ | |
4 | leadd1 7425 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))) | |
5 | 1, 2, 3, 4 | mp3an 1232 | 1 ⊢ (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 ℝcr 6888 + caddc 6892 ≤ cle 7061 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-iota 4867 df-fv 4910 df-ov 5515 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |