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Mirrors > Home > MPE Home > Th. List > leadd2dd | Structured version Visualization version GIF version |
Description: Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
leadd1dd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
leadd2dd | ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leadd1dd.4 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
2 | leidd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltnegd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | ltadd1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 2, 3, 4 | leadd2d 10501 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 221 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 + caddc 9818 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: difgtsumgt 11223 expmulnbnd 12858 discr1 12862 hashun2 13033 abstri 13918 iseraltlem2 14261 prmreclem4 15461 tchcphlem1 22842 trirn 22991 nulmbl2 23111 voliunlem1 23125 uniioombllem4 23160 itg2split 23322 ulmcn 23957 abslogle 24168 emcllem2 24523 lgambdd 24563 chtublem 24736 chtub 24737 logfaclbnd 24747 bcmax 24803 chebbnd1lem2 24959 rplogsumlem1 24973 selberglem2 25035 selbergb 25038 chpdifbndlem1 25042 pntpbnd1a 25074 pntpbnd2 25076 pntibndlem2 25080 pntibndlem3 25081 pntlemg 25087 pntlemr 25091 pntlemk 25095 pntlemo 25096 ostth2lem3 25124 smcnlem 26936 minvecolem3 27116 staddi 28489 stadd3i 28491 nexple 29399 rescon 30482 itg2addnc 32634 ftc1anclem8 32662 pell1qrgaplem 36455 leadd12dd 38473 ioodvbdlimc1lem2 38822 stoweidlem11 38904 stoweidlem26 38919 stirlinglem8 38974 stirlinglem12 38978 fourierdlem4 39004 fourierdlem10 39010 fourierdlem42 39042 fourierdlem47 39046 fourierdlem72 39071 fourierdlem79 39078 fourierdlem93 39092 fourierdlem101 39100 fourierdlem103 39102 fourierdlem104 39103 fourierdlem111 39110 hoidmv1lelem2 39482 vonioolem2 39572 vonicclem2 39575 fmtnodvds 39994 lighneallem4a 40063 p1lep2 40346 |
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