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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 13072 | . 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 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 + caddc 11158 < clt 11295 ℝ+crp 13034 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-rp 13035 |
| This theorem is referenced by: ltaddrp2d 13111 xov1plusxeqvd 13538 isumltss 15884 effsumlt 16147 tanhlt1 16196 4sqlem12 16994 vdwlem1 17019 prmgaplem7 17095 chfacfscmul0 22864 chfacfpmmul0 22868 nlmvscnlem2 24706 nlmvscnlem1 24707 iccntr 24843 icccmplem2 24845 reconnlem2 24849 opnreen 24853 lebnumii 24998 ipcnlem2 25278 ipcnlem1 25279 ivthlem2 25487 ovolgelb 25515 ovollb2lem 25523 itg2monolem3 25787 dvferm1lem 26022 lhop1lem 26052 lhop 26055 dvcnvrelem1 26056 dvcnvrelem2 26057 pserdvlem1 26471 pserdv 26473 lgamgulmlem2 27073 lgamgulmlem3 27074 lgamucov 27081 perfectlem2 27274 bposlem2 27329 pntibndlem2 27635 pntlemb 27641 pntlem3 27653 tpr2rico 33911 omssubaddlem 34301 fibp1 34403 heicant 37662 itg2addnc 37681 rrnequiv 37842 2np3bcnp1 42145 2ap1caineq 42146 pellfundex 42897 rmspecfund 42920 acongeq 42995 jm3.1lem2 43030 oddfl 45289 infrpge 45362 xralrple2 45365 xrralrecnnle 45394 iooiinicc 45555 iooiinioc 45569 fsumnncl 45587 climinf 45621 lptre2pt 45655 ioodvbdlimc1lem2 45947 wallispilem4 46083 dirkertrigeqlem3 46115 dirkercncflem2 46119 fourierdlem63 46184 fourierdlem65 46186 fourierdlem75 46196 fourierdlem79 46200 fouriersw 46246 etransclem35 46284 qndenserrnbllem 46309 omeiunltfirp 46534 hoidmvlelem1 46610 hoidmvlelem3 46612 hoiqssbllem3 46639 iinhoiicc 46689 iunhoiioo 46691 vonioolem2 46696 vonicclem1 46698 preimaleiinlt 46736 smfmullem3 46808 perfectALTVlem2 47709 |
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