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Mirrors > Home > ILE Home > Th. List > le2addd | GIF version |
Description: Adding both side of two inequalities. (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) |
le2addd.5 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
le2addd.6 | ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
Ref | Expression |
---|---|
le2addd | ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | le2addd.5 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
2 | le2addd.6 | . 2 ⊢ (𝜑 → 𝐵 ≤ 𝐷) | |
3 | leidd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | ltnegd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | ltadd1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | lt2addd.4 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | le2add 8388 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | |
8 | 3, 4, 5, 6, 7 | syl22anc 1239 | . 2 ⊢ (𝜑 → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) |
9 | 1, 2, 8 | mp2and 433 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5869 ℝcr 7798 + caddc 7802 ≤ cle 7980 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-i2m1 7904 ax-0id 7907 ax-rnegex 7908 ax-pre-ltwlin 7912 ax-pre-ltadd 7915 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-iota 5174 df-fv 5220 df-ov 5872 df-pnf 7981 df-mnf 7982 df-xr 7983 df-ltxr 7984 df-le 7985 |
This theorem is referenced by: lgsdirprm 14095 2sqlem8 14119 |
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