Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12416 | . 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 2105 class class class wbr 5058 (class class class)co 7145 ℝcr 10525 + caddc 10529 < clt 10664 ℝ+crp 12379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-rp 12380 |
This theorem is referenced by: ltaddrp2d 12455 xov1plusxeqvd 12874 isumltss 15193 effsumlt 15454 tanhlt1 15503 4sqlem12 16282 vdwlem1 16307 prmgaplem7 16383 chfacfscmul0 21396 chfacfpmmul0 21400 nlmvscnlem2 23223 nlmvscnlem1 23224 iccntr 23358 icccmplem2 23360 reconnlem2 23364 opnreen 23368 lebnumii 23499 ipcnlem2 23776 ipcnlem1 23777 ivthlem2 23982 ovolgelb 24010 ovollb2lem 24018 itg2monolem3 24282 dvferm1lem 24510 lhop1lem 24539 lhop 24542 dvcnvrelem1 24543 dvcnvrelem2 24544 pserdvlem1 24944 pserdv 24946 lgamgulmlem2 25535 lgamgulmlem3 25536 lgamucov 25543 perfectlem2 25734 bposlem2 25789 pntibndlem2 26095 pntlemb 26101 pntlem3 26113 tpr2rico 31055 omssubaddlem 31457 fibp1 31559 heicant 34809 itg2addnc 34828 rrnequiv 34996 pellfundex 39363 rmspecfund 39386 acongeq 39460 jm3.1lem2 39495 oddfl 41423 infrpge 41499 xralrple2 41502 xrralrecnnle 41533 iooiinicc 41698 iooiinioc 41712 fsumnncl 41732 climinf 41767 lptre2pt 41801 ioodvbdlimc1lem2 42097 wallispilem4 42234 dirkertrigeqlem3 42266 dirkercncflem2 42270 fourierdlem63 42335 fourierdlem65 42337 fourierdlem75 42347 fourierdlem79 42351 fouriersw 42397 etransclem35 42435 qndenserrnbllem 42460 omeiunltfirp 42682 hoidmvlelem1 42758 hoidmvlelem3 42760 hoiqssbllem3 42787 iinhoiicc 42837 iunhoiioo 42839 vonioolem2 42844 vonicclem1 42846 preimaleiinlt 42880 smfmullem3 42949 perfectALTVlem2 43734 |
Copyright terms: Public domain | W3C validator |