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Mirrors > Home > MPE Home > Th. List > le2addi | Structured version Visualization version GIF version |
Description: Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.) |
Ref | Expression |
---|---|
lt2.1 | ⊢ 𝐴 ∈ ℝ |
lt2.2 | ⊢ 𝐵 ∈ ℝ |
lt2.3 | ⊢ 𝐶 ∈ ℝ |
lt.4 | ⊢ 𝐷 ∈ ℝ |
Ref | Expression |
---|---|
le2addi | ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | lt2.2 | . 2 ⊢ 𝐵 ∈ ℝ | |
3 | lt2.3 | . 2 ⊢ 𝐶 ∈ ℝ | |
4 | lt.4 | . 2 ⊢ 𝐷 ∈ ℝ | |
5 | le2add 11668 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷))) | |
6 | 1, 2, 3, 4, 5 | mp4an 691 | 1 ⊢ ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5132 (class class class)co 7384 ℝcr 11081 + caddc 11085 ≤ cle 11221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-br 5133 df-opab 5195 df-mpt 5216 df-id 5558 df-po 5572 df-so 5573 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-ov 7387 df-er 8677 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 |
This theorem is referenced by: hashunlei 14357 log2ublem1 26355 unierri 31150 ballotlem2 33218 |
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