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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 12623 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵)) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → 𝐴 < (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 + caddc 10732 < clt 10867 ℝ+crp 12586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-ltxr 10872 df-rp 12587 |
This theorem is referenced by: ltaddrp2d 12662 xov1plusxeqvd 13086 isumltss 15412 effsumlt 15672 tanhlt1 15721 4sqlem12 16509 vdwlem1 16534 prmgaplem7 16610 chfacfscmul0 21755 chfacfpmmul0 21759 nlmvscnlem2 23583 nlmvscnlem1 23584 iccntr 23718 icccmplem2 23720 reconnlem2 23724 opnreen 23728 lebnumii 23863 ipcnlem2 24141 ipcnlem1 24142 ivthlem2 24349 ovolgelb 24377 ovollb2lem 24385 itg2monolem3 24650 dvferm1lem 24881 lhop1lem 24910 lhop 24913 dvcnvrelem1 24914 dvcnvrelem2 24915 pserdvlem1 25319 pserdv 25321 lgamgulmlem2 25912 lgamgulmlem3 25913 lgamucov 25920 perfectlem2 26111 bposlem2 26166 pntibndlem2 26472 pntlemb 26478 pntlem3 26490 tpr2rico 31576 omssubaddlem 31978 fibp1 32080 heicant 35549 itg2addnc 35568 rrnequiv 35730 2np3bcnp1 39822 2ap1caineq 39823 pellfundex 40411 rmspecfund 40434 acongeq 40508 jm3.1lem2 40543 oddfl 42488 infrpge 42563 xralrple2 42566 xrralrecnnle 42595 iooiinicc 42755 iooiinioc 42769 fsumnncl 42788 climinf 42822 lptre2pt 42856 ioodvbdlimc1lem2 43148 wallispilem4 43284 dirkertrigeqlem3 43316 dirkercncflem2 43320 fourierdlem63 43385 fourierdlem65 43387 fourierdlem75 43397 fourierdlem79 43401 fouriersw 43447 etransclem35 43485 qndenserrnbllem 43510 omeiunltfirp 43732 hoidmvlelem1 43808 hoidmvlelem3 43810 hoiqssbllem3 43837 iinhoiicc 43887 iunhoiioo 43889 vonioolem2 43894 vonicclem1 43896 preimaleiinlt 43930 smfmullem3 43999 perfectALTVlem2 44847 |
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