Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leadd12dd | Structured version Visualization version GIF version |
Description: Addition to both sides of 'less than or equal to'. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
leadd12dd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
leadd12dd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
leadd12dd.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
leadd12dd.d | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
leadd12dd.ac | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
leadd12dd.bd | ⊢ (𝜑 → 𝐵 ≤ 𝐷) |
Ref | Expression |
---|---|
leadd12dd | ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leadd12dd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | leadd12dd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 11014 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | leadd12dd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 4, 2 | readdcld 11014 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) ∈ ℝ) |
6 | leadd12dd.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
7 | 4, 6 | readdcld 11014 | . 2 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℝ) |
8 | leadd12dd.ac | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
9 | 1, 4, 2, 8 | leadd1dd 11599 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐵)) |
10 | leadd12dd.bd | . . 3 ⊢ (𝜑 → 𝐵 ≤ 𝐷) | |
11 | 2, 6, 4, 10 | leadd2dd 11600 | . 2 ⊢ (𝜑 → (𝐶 + 𝐵) ≤ (𝐶 + 𝐷)) |
12 | 3, 5, 7, 9, 11 | letrd 11142 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5073 (class class class)co 7267 ℝcr 10880 + caddc 10884 ≤ cle 11020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-po 5498 df-so 5499 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-ov 7270 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 |
This theorem is referenced by: sge0xaddlem1 43952 hoidmvlelem2 44115 hspmbllem2 44146 smfmullem1 44303 |
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