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| Mirrors > Home > MPE Home > Th. List > lesub1dd | Structured version Visualization version GIF version | ||
| Description: Subtraction from 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 | 
|---|---|
| lesub1dd | ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) | 
| 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 | lesub1d 11871 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))) | 
| 6 | 1, 5 | mpbid 232 | 1 ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 ≤ cle 11297 − cmin 11493 | 
| 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-reu 3380 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-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 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 df-sub 11495 df-neg 11496 | 
| This theorem is referenced by: eluzmn 12886 elfzmlbm 13679 modmulnn 13930 icodiamlt 15475 rlimrege0 15616 climsqz2 15679 rlimsqz2 15688 isercolllem1 15702 caucvgrlem 15710 climcndslem1 15886 bitsinv1lem 16479 hashdvds 16813 4sqlem6 16982 dvfsumlem2 26068 dvfsumlem2OLD 26069 dvfsumlem4 26071 dvfsum2 26076 isosctrlem1 26862 lgamgulmlem2 27074 basellem9 27133 ppiub 27249 chtub 27257 logfaclbnd 27267 bposlem1 27329 bposlem6 27334 selberg2lem 27595 pntpbnd2 27632 pntlemo 27652 ttgcontlem1 28900 axpaschlem 28956 axcontlem8 28987 cycpmco2lem7 33153 dnibndlem10 36489 unblimceq0 36509 unbdqndv2lem2 36512 poimirlem6 37634 poimirlem7 37635 itg2addnclem3 37681 iccbnd 37848 lcmineqlem23 42053 sticksstones12a 42159 sticksstones12 42160 bcled 42180 bcle2d 42181 metakunt30 42236 jm2.24nn 42976 fzmaxdif 42998 areaquad 43233 monoords 45314 iccshift 45536 climinf 45626 sumnnodd 45650 dvnmul 45963 itgiccshift 46000 itgperiod 46001 itgsbtaddcnst 46002 stoweidlem13 46033 stoweidlem26 46046 stoweidlem34 46054 fourierdlem19 46146 fourierdlem42 46169 fourierdlem74 46200 fourierdlem75 46201 fourierdlem79 46205 fourierdlem81 46207 fourierdlem82 46208 fourierdlem103 46229 fourierdlem104 46230 fouriersw 46251 hoidmvlelem1 46615 bgoldbtbndlem2 47798 | 
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