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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 11859 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))) | 
| 6 | 1, 5 | mpbid 232 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 + caddc 11159 ≤ cle 11297 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 | 
| This theorem is referenced by: difgtsumgt 12581 expmulnbnd 14275 discr1 14279 hashun2 14423 abstri 15370 iseraltlem2 15720 prmreclem4 16958 tcphcphlem1 25270 trirn 25435 nulmbl2 25572 voliunlem1 25586 uniioombllem4 25622 itg2split 25785 ulmcn 26443 abslogle 26661 emcllem2 27041 lgambdd 27081 chtublem 27256 chtub 27257 logfaclbnd 27267 bcmax 27323 chebbnd1lem2 27515 rplogsumlem1 27529 selberglem2 27591 selbergb 27594 chpdifbndlem1 27598 pntpbnd1a 27630 pntpbnd2 27632 pntibndlem2 27636 pntibndlem3 27637 pntlemg 27643 pntlemr 27647 pntlemk 27651 pntlemo 27652 ostth2lem3 27680 smcnlem 30717 minvecolem3 30896 staddi 32266 stadd3i 32268 nexple 32834 fsum2dsub 34623 resconn 35252 itg2addnc 37682 ftc1anclem8 37708 lcmineqlem22 42052 aks4d1p1p2 42072 aks4d1p1p5 42077 bcle2d 42181 aks6d1c7lem1 42182 fimgmcyc 42549 pell1qrgaplem 42889 leadd12dd 45333 ioodvbdlimc1lem2 45952 stoweidlem11 46031 stoweidlem26 46046 stirlinglem8 46101 stirlinglem12 46105 fourierdlem4 46131 fourierdlem10 46137 fourierdlem42 46169 fourierdlem47 46173 fourierdlem72 46198 fourierdlem79 46205 fourierdlem93 46219 fourierdlem101 46227 fourierdlem103 46229 fourierdlem104 46230 fourierdlem111 46237 hoidmv1lelem2 46612 vonioolem2 46701 vonicclem2 46704 p1lep2 47317 fmtnodvds 47536 lighneallem4a 47600 | 
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