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Mirrors > Home > MPE Home > Th. List > lesubaddd | Structured version Visualization version GIF version |
Description: 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
lesubaddd | ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltnegd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltadd1d.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | lesubadd 11586 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 class class class wbr 5104 (class class class)co 7352 ℝcr 11009 + caddc 11013 ≤ cle 11149 − cmin 11344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 |
This theorem is referenced by: elfzomelpfzo 13631 modaddmodup 13794 01sqrexlem7 15093 absrdbnd 15186 caucvgrlem 15517 cvgcmp 15661 oddge22np1 16191 ramub1lem1 16858 chfacfisf 22155 chfacfisfcpmat 22156 uniioombllem4 24902 mbfi1fseqlem6 25037 dvfsumlem1 25342 abelthlem2 25743 argimgt0 25919 harmonicbnd4 26312 ppiub 26504 logfaclbnd 26522 logfacbnd3 26523 bcmax 26578 lgseisen 26679 log2sumbnd 26844 chpdifbndlem1 26853 pntpbnd2 26887 pntibndlem2 26891 pntlemo 26907 crctcshwlkn0lem5 28588 clwlkclwwlklem2 28773 clwlkclwwlk2 28776 nvabs 29443 dnibndlem4 34876 dnibndlem10 34882 itg2addnclem2 36062 itg2addnclem3 36063 metakunt16 40524 fzmaxdif 41208 int-ineqmvtd 42369 binomcxplemnotnn0 42541 xrralrecnnge 43524 limsupgtlem 43913 fourierdlem26 44269 hoidmv1lelem1 44727 leaddsuble 45424 fmtnoge3 45617 fmtnoprmfac2lem1 45653 bgoldbtbndlem2 45893 nnolog2flm1 46571 |
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