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Mirrors > Home > MPE Home > Th. List > ltsubrpd | Structured version Visualization version GIF version |
Description: Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rpgecld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
ltsubrpd | ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rpgecld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | ltsubrp 11904 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴) | |
4 | 1, 2, 3 | syl2anc 694 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 class class class wbr 4685 (class class class)co 6690 ℝcr 9973 < clt 10112 − cmin 10304 ℝ+crp 11870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-neg 10307 df-rp 11871 |
This theorem is referenced by: tanhlt1 14934 pythagtriplem13 15579 iccntr 22671 icccmplem2 22673 opnreen 22681 evth 22805 ovollb2lem 23302 ismbf3d 23466 itg2seq 23554 itg2cn 23575 dvferm2lem 23794 lhop 23824 dvcnvrelem1 23825 dvcnvrelem2 23826 aaliou3lem7 24149 lgseisenlem1 25145 pntlem3 25343 lt2addrd 29644 ltesubnnd 29696 tpr2rico 30086 fiblem 30588 signstfveq0 30782 mblfinlem3 33578 mblfinlem4 33579 suprltrp 39857 suplesup 39868 xrralrecnnge 39926 iooiinicc 40087 sumnnodd 40180 lptre2pt 40190 ioodvbdlimc2lem 40467 dvnmul 40476 stoweidlem18 40553 fourierdlem107 40748 fouriersw 40766 hoiqssbllem3 41159 ovolval5lem2 41188 preimageiingt 41251 smfmullem3 41321 |
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