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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 11249 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))) |
6 | 1, 5 | mpbid 234 | 1 ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 ≤ cle 10678 − cmin 10872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 |
This theorem is referenced by: eluzmn 12253 elfzmlbm 13020 modmulnn 13260 icodiamlt 14797 rlimrege0 14938 climsqz2 15000 rlimsqz2 15009 isercolllem1 15023 caucvgrlem 15031 climcndslem1 15206 bitsinv1lem 15792 hashdvds 16114 4sqlem6 16281 dvfsumlem2 24626 dvfsumlem4 24628 dvfsum2 24633 isosctrlem1 25398 lgamgulmlem2 25609 basellem9 25668 ppiub 25782 chtub 25790 logfaclbnd 25800 bposlem1 25862 bposlem6 25867 selberg2lem 26128 pntpbnd2 26165 pntlemo 26185 ttgcontlem1 26673 axpaschlem 26728 axcontlem8 26759 cycpmco2lem7 30776 dnibndlem10 33828 unblimceq0 33848 unbdqndv2lem2 33851 poimirlem6 34900 poimirlem7 34901 itg2addnclem3 34947 iccbnd 35120 jm2.24nn 39563 fzmaxdif 39585 areaquad 39830 monoords 41571 iccshift 41801 climinf 41894 sumnnodd 41918 dvnmul 42235 itgiccshift 42272 itgperiod 42273 itgsbtaddcnst 42274 stoweidlem13 42305 stoweidlem26 42318 stoweidlem34 42326 fourierdlem19 42418 fourierdlem42 42441 fourierdlem74 42472 fourierdlem75 42473 fourierdlem79 42477 fourierdlem81 42479 fourierdlem82 42480 fourierdlem103 42501 fourierdlem104 42502 fouriersw 42523 hoidmvlelem1 42884 bgoldbtbndlem2 43978 |
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