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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 11817 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶))) |
6 | 1, 5 | mpbid 231 | 1 ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5147 (class class class)co 7405 ℝcr 11105 ≤ cle 11245 − cmin 11440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 |
This theorem is referenced by: eluzmn 12825 elfzmlbm 13607 modmulnn 13850 icodiamlt 15378 rlimrege0 15519 climsqz2 15582 rlimsqz2 15593 isercolllem1 15607 caucvgrlem 15615 climcndslem1 15791 bitsinv1lem 16378 hashdvds 16704 4sqlem6 16872 dvfsumlem2 25535 dvfsumlem4 25537 dvfsum2 25542 isosctrlem1 26312 lgamgulmlem2 26523 basellem9 26582 ppiub 26696 chtub 26704 logfaclbnd 26714 bposlem1 26776 bposlem6 26781 selberg2lem 27042 pntpbnd2 27079 pntlemo 27099 ttgcontlem1 28131 axpaschlem 28187 axcontlem8 28218 cycpmco2lem7 32278 gg-dvfsumlem2 35171 dnibndlem10 35351 unblimceq0 35371 unbdqndv2lem2 35374 poimirlem6 36482 poimirlem7 36483 itg2addnclem3 36529 iccbnd 36696 lcmineqlem23 40904 sticksstones12a 40961 sticksstones12 40962 metakunt30 41002 jm2.24nn 41683 fzmaxdif 41705 areaquad 41950 monoords 43993 iccshift 44217 climinf 44308 sumnnodd 44332 dvnmul 44645 itgiccshift 44682 itgperiod 44683 itgsbtaddcnst 44684 stoweidlem13 44715 stoweidlem26 44728 stoweidlem34 44736 fourierdlem19 44828 fourierdlem42 44851 fourierdlem74 44882 fourierdlem75 44883 fourierdlem79 44887 fourierdlem81 44889 fourierdlem82 44890 fourierdlem103 44911 fourierdlem104 44912 fouriersw 44933 hoidmvlelem1 45297 bgoldbtbndlem2 46460 |
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