Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11247 | . 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 5066 (class class class)co 7156 ℝcr 10536 ≤ cle 10676 − cmin 10870 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 |
This theorem is referenced by: eluzmn 12251 elfzmlbm 13018 modmulnn 13258 icodiamlt 14795 rlimrege0 14936 climsqz2 14998 rlimsqz2 15007 isercolllem1 15021 caucvgrlem 15029 climcndslem1 15204 bitsinv1lem 15790 hashdvds 16112 4sqlem6 16279 dvfsumlem2 24624 dvfsumlem4 24626 dvfsum2 24631 isosctrlem1 25396 lgamgulmlem2 25607 basellem9 25666 ppiub 25780 chtub 25788 logfaclbnd 25798 bposlem1 25860 bposlem6 25865 selberg2lem 26126 pntpbnd2 26163 pntlemo 26183 ttgcontlem1 26671 axpaschlem 26726 axcontlem8 26757 cycpmco2lem7 30774 dnibndlem10 33826 unblimceq0 33846 unbdqndv2lem2 33849 poimirlem6 34913 poimirlem7 34914 itg2addnclem3 34960 iccbnd 35133 jm2.24nn 39605 fzmaxdif 39627 areaquad 39872 monoords 41613 iccshift 41843 climinf 41936 sumnnodd 41960 dvnmul 42277 itgiccshift 42314 itgperiod 42315 itgsbtaddcnst 42316 stoweidlem13 42347 stoweidlem26 42360 stoweidlem34 42368 fourierdlem19 42460 fourierdlem42 42483 fourierdlem74 42514 fourierdlem75 42515 fourierdlem79 42519 fourierdlem81 42521 fourierdlem82 42522 fourierdlem103 42543 fourierdlem104 42544 fouriersw 42565 hoidmvlelem1 42926 bgoldbtbndlem2 44020 |
Copyright terms: Public domain | W3C validator |