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Mirrors > Home > MPE Home > Th. List > leaddsub | Structured version Visualization version GIF version |
Description: 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.) |
Ref | Expression |
---|---|
leaddsub | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsubadd 11201 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 − 𝐵) < 𝐴 ↔ 𝐶 < (𝐴 + 𝐵))) | |
2 | 1 | 3com13 1125 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) < 𝐴 ↔ 𝐶 < (𝐴 + 𝐵))) |
3 | resubcl 11041 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 − 𝐵) ∈ ℝ) | |
4 | ltnle 10811 | . . . . 5 ⊢ (((𝐶 − 𝐵) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 − 𝐵) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐶 − 𝐵))) | |
5 | 3, 4 | stoic3 1783 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐶 − 𝐵) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐶 − 𝐵))) |
6 | 5 | 3com13 1125 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐶 − 𝐵) < 𝐴 ↔ ¬ 𝐴 ≤ (𝐶 − 𝐵))) |
7 | readdcl 10711 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
8 | ltnle 10811 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ (𝐴 + 𝐵) ∈ ℝ) → (𝐶 < (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) ≤ 𝐶)) | |
9 | 7, 8 | sylan2 596 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐶 < (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) ≤ 𝐶)) |
10 | 9 | 3impb 1116 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 < (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) ≤ 𝐶)) |
11 | 10 | 3coml 1128 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < (𝐴 + 𝐵) ↔ ¬ (𝐴 + 𝐵) ≤ 𝐶)) |
12 | 2, 6, 11 | 3bitr3rd 313 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (¬ (𝐴 + 𝐵) ≤ 𝐶 ↔ ¬ 𝐴 ≤ (𝐶 − 𝐵))) |
13 | 12 | con4bid 320 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2114 class class class wbr 5040 (class class class)co 7183 ℝcr 10627 + caddc 10631 < clt 10766 ≤ cle 10767 − cmin 10961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 |
This theorem is referenced by: leaddsub2 11208 lesub 11210 lesub2 11226 subge0 11244 lesub3d 11349 div4p1lem1div2 11984 eluzp1m1 12363 eluzsubi 12367 fzen 13028 fznatpl1 13065 expmulnbnd 13701 hashdvds 16225 sylow1lem5 18858 gsumbagdiaglemOLD 20764 gsumbagdiaglem 20767 voliunlem2 24316 itg2split 24515 dvfsumlem3 24793 pilem2 25212 logimul 25370 emcllem2 25747 chtublem 25960 dchrisum0re 26262 pntlemg 26347 crctcshwlkn0 27772 logdivsqrle 32213 poimirlem7 35440 totbndbnd 35603 aks4d1p1p5 39735 aks4d1p1 39736 sticksstones10 39750 sticksstones12a 39752 sticksstones12 39753 metakunt16 39772 binomcxplemnn0 41546 fmtnodvds 44578 lighneallem4a 44642 nnolog2flm1 45518 |
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