![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 10660 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 222 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 class class class wbr 4685 (class class class)co 6690 ℝcr 9973 + caddc 9977 ≤ cle 10113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 |
This theorem is referenced by: difgtsumgt 11384 expmulnbnd 13036 discr1 13040 hashun2 13210 abstri 14114 iseraltlem2 14457 prmreclem4 15670 tchcphlem1 23080 trirn 23229 nulmbl2 23350 voliunlem1 23364 uniioombllem4 23400 itg2split 23561 ulmcn 24198 abslogle 24409 emcllem2 24768 lgambdd 24808 chtublem 24981 chtub 24982 logfaclbnd 24992 bcmax 25048 chebbnd1lem2 25204 rplogsumlem1 25218 selberglem2 25280 selbergb 25283 chpdifbndlem1 25287 pntpbnd1a 25319 pntpbnd2 25321 pntibndlem2 25325 pntibndlem3 25326 pntlemg 25332 pntlemr 25336 pntlemk 25340 pntlemo 25341 ostth2lem3 25369 smcnlem 27680 minvecolem3 27860 staddi 29233 stadd3i 29235 nexple 30199 fsum2dsub 30813 resconn 31354 itg2addnc 33594 ftc1anclem8 33622 pell1qrgaplem 37754 leadd12dd 39845 ioodvbdlimc1lem2 40465 stoweidlem11 40546 stoweidlem26 40561 stirlinglem8 40616 stirlinglem12 40620 fourierdlem4 40646 fourierdlem10 40652 fourierdlem42 40684 fourierdlem47 40688 fourierdlem72 40713 fourierdlem79 40720 fourierdlem93 40734 fourierdlem101 40742 fourierdlem103 40744 fourierdlem104 40745 fourierdlem111 40752 hoidmv1lelem2 41127 vonioolem2 41216 vonicclem2 41219 p1lep2 41639 fmtnodvds 41781 lighneallem4a 41850 |
Copyright terms: Public domain | W3C validator |