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Mirrors > Home > ILE Home > Th. List > lt2addd | GIF version |
Description: Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
lt2addd.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
lt2addd.5 | ⊢ (𝜑 → 𝐴 < 𝐶) |
lt2addd.6 | ⊢ (𝜑 → 𝐵 < 𝐷) |
Ref | Expression |
---|---|
lt2addd | ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltadd1d.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lt2addd.4 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
5 | lt2addd.5 | . 2 ⊢ (𝜑 → 𝐴 < 𝐶) | |
6 | lt2addd.6 | . . 3 ⊢ (𝜑 → 𝐵 < 𝐷) | |
7 | 2, 4, 6 | ltled 7849 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
8 | 1, 2, 3, 4, 5, 7 | ltleaddd 8295 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < (𝐶 + 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 ℝcr 7587 + caddc 7591 < clt 7768 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-iota 5058 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: modaddmodup 10128 modsumfzodifsn 10137 resqrexlemnm 10758 mertenslemi1 11272 |
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