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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 11114 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 + caddc 10542 ≤ 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: elfzomelpfzo 13144 modaddmodup 13305 sqrlem7 14610 absrdbnd 14703 caucvgrlem 15031 cvgcmp 15173 oddge22np1 15700 ramub1lem1 16364 chfacfisf 21464 chfacfisfcpmat 21465 uniioombllem4 24189 mbfi1fseqlem6 24323 dvfsumlem1 24625 abelthlem2 25022 argimgt0 25197 harmonicbnd4 25590 ppiub 25782 logfaclbnd 25800 logfacbnd3 25801 bcmax 25856 lgseisen 25957 log2sumbnd 26122 chpdifbndlem1 26131 pntpbnd2 26165 pntibndlem2 26169 pntlemo 26185 crctcshwlkn0lem5 27594 clwlkclwwlklem2 27780 clwlkclwwlk2 27783 nvabs 28451 dnibndlem4 33822 dnibndlem10 33828 itg2addnclem2 34946 itg2addnclem3 34947 fzmaxdif 39585 int-ineqmvtd 40551 binomcxplemnotnn0 40695 xrralrecnnge 41669 limsupgtlem 42065 fourierdlem26 42425 hoidmv1lelem1 42880 leaddsuble 43504 fmtnoge3 43699 fmtnoprmfac2lem1 43735 bgoldbtbndlem2 43978 nnolog2flm1 44657 |
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