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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 11866 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 − 𝐵) < 𝐴) | |
4 | 1, 2, 3 | syl2anc 693 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 < clt 10074 − cmin 10266 ℝ+crp 11832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-rp 11833 |
This theorem is referenced by: tanhlt1 14890 pythagtriplem13 15532 iccntr 22624 icccmplem2 22626 opnreen 22634 evth 22758 ovollb2lem 23256 ismbf3d 23421 itg2seq 23509 itg2cn 23530 dvferm2lem 23749 lhop 23779 dvcnvrelem1 23780 dvcnvrelem2 23781 aaliou3lem7 24104 lgseisenlem1 25100 pntlem3 25298 lt2addrd 29516 ltesubnnd 29568 tpr2rico 29958 fiblem 30460 signstfveq0 30654 mblfinlem3 33448 mblfinlem4 33449 suprltrp 39544 suplesup 39555 xrralrecnnge 39613 iooiinicc 39769 sumnnodd 39862 lptre2pt 39872 ioodvbdlimc2lem 40149 dvnmul 40158 stoweidlem18 40235 fourierdlem107 40430 fouriersw 40448 hoiqssbllem3 40838 ovolval5lem2 40867 preimageiingt 40930 smfmullem3 41000 |
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