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Mirrors > Home > MPE Home > Th. List > ltaddrpd | Structured version Visualization version GIF version |
Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
rpgecld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
Ref | Expression |
---|---|
ltaddrpd | ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rpgecld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
3 | ltaddrp 12993 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵)) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5141 (class class class)co 7393 ℝcr 11091 + caddc 11095 < clt 11230 ℝ+crp 12956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 df-rp 12957 |
This theorem is referenced by: ltaddrp2d 13032 xov1plusxeqvd 13457 isumltss 15776 effsumlt 16036 tanhlt1 16085 4sqlem12 16871 vdwlem1 16896 prmgaplem7 16972 chfacfscmul0 22289 chfacfpmmul0 22293 nlmvscnlem2 24131 nlmvscnlem1 24132 iccntr 24266 icccmplem2 24268 reconnlem2 24272 opnreen 24276 lebnumii 24411 ipcnlem2 24690 ipcnlem1 24691 ivthlem2 24898 ovolgelb 24926 ovollb2lem 24934 itg2monolem3 25199 dvferm1lem 25430 lhop1lem 25459 lhop 25462 dvcnvrelem1 25463 dvcnvrelem2 25464 pserdvlem1 25868 pserdv 25870 lgamgulmlem2 26461 lgamgulmlem3 26462 lgamucov 26469 perfectlem2 26660 bposlem2 26715 pntibndlem2 27021 pntlemb 27027 pntlem3 27039 tpr2rico 32723 omssubaddlem 33129 fibp1 33231 heicant 36327 itg2addnc 36346 rrnequiv 36508 2np3bcnp1 40765 2ap1caineq 40766 pellfundex 41395 rmspecfund 41418 acongeq 41493 jm3.1lem2 41528 oddfl 43760 infrpge 43834 xralrple2 43837 xrralrecnnle 43866 iooiinicc 44028 iooiinioc 44042 fsumnncl 44061 climinf 44095 lptre2pt 44129 ioodvbdlimc1lem2 44421 wallispilem4 44557 dirkertrigeqlem3 44589 dirkercncflem2 44593 fourierdlem63 44658 fourierdlem65 44660 fourierdlem75 44670 fourierdlem79 44674 fouriersw 44720 etransclem35 44758 qndenserrnbllem 44783 omeiunltfirp 45008 hoidmvlelem1 45084 hoidmvlelem3 45086 hoiqssbllem3 45113 iinhoiicc 45163 iunhoiioo 45165 vonioolem2 45170 vonicclem1 45172 preimaleiinlt 45210 smfmullem3 45282 perfectALTVlem2 46162 |
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