Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11304 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 + caddc 10732 ≤ cle 10868 − cmin 11062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 |
This theorem is referenced by: elfzomelpfzo 13346 modaddmodup 13507 sqrlem7 14812 absrdbnd 14905 caucvgrlem 15236 cvgcmp 15380 oddge22np1 15910 ramub1lem1 16579 chfacfisf 21751 chfacfisfcpmat 21752 uniioombllem4 24483 mbfi1fseqlem6 24618 dvfsumlem1 24923 abelthlem2 25324 argimgt0 25500 harmonicbnd4 25893 ppiub 26085 logfaclbnd 26103 logfacbnd3 26104 bcmax 26159 lgseisen 26260 log2sumbnd 26425 chpdifbndlem1 26434 pntpbnd2 26468 pntibndlem2 26472 pntlemo 26488 crctcshwlkn0lem5 27898 clwlkclwwlklem2 28083 clwlkclwwlk2 28086 nvabs 28753 dnibndlem4 34398 dnibndlem10 34404 itg2addnclem2 35566 itg2addnclem3 35567 metakunt16 39862 fzmaxdif 40506 int-ineqmvtd 41480 binomcxplemnotnn0 41647 xrralrecnnge 42603 limsupgtlem 42993 fourierdlem26 43349 hoidmv1lelem1 43804 leaddsuble 44462 fmtnoge3 44655 fmtnoprmfac2lem1 44691 bgoldbtbndlem2 44931 nnolog2flm1 45609 |
Copyright terms: Public domain | W3C validator |