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Mirrors > Home > ILE Home > Th. List > addlelt | GIF version |
Description: If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.) |
Ref | Expression |
---|---|
addlelt | ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 + 𝐴) ≤ 𝑁 → 𝑀 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgt0 9446 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
2 | 1 | 3ad2ant3 1004 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 0 < 𝐴) |
3 | rpre 9441 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
4 | 3 | 3ad2ant3 1004 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
5 | simp1 981 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 𝑀 ∈ ℝ) | |
6 | 4, 5 | ltaddposd 8284 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → (0 < 𝐴 ↔ 𝑀 < (𝑀 + 𝐴))) |
7 | 2, 6 | mpbid 146 | . 2 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 𝑀 < (𝑀 + 𝐴)) |
8 | simpl 108 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 𝑀 ∈ ℝ) | |
9 | 3 | adantl 275 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
10 | 8, 9 | readdcld 7788 | . . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → (𝑀 + 𝐴) ∈ ℝ) |
11 | 10 | 3adant2 1000 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → (𝑀 + 𝐴) ∈ ℝ) |
12 | simp2 982 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 𝑁 ∈ ℝ) | |
13 | ltletr 7846 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 𝐴) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 < (𝑀 + 𝐴) ∧ (𝑀 + 𝐴) ≤ 𝑁) → 𝑀 < 𝑁)) | |
14 | 5, 11, 12, 13 | syl3anc 1216 | . 2 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 < (𝑀 + 𝐴) ∧ (𝑀 + 𝐴) ≤ 𝑁) → 𝑀 < 𝑁)) |
15 | 7, 14 | mpand 425 | 1 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → ((𝑀 + 𝐴) ≤ 𝑁 → 𝑀 < 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 0cc0 7613 + caddc 7616 < clt 7793 ≤ cle 7794 ℝ+crp 9434 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-pre-ltwlin 7726 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-rp 9435 |
This theorem is referenced by: zltaddlt1le 9782 |
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