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| Mirrors > Home > MPE Home > Th. List > ge0p1rpd | Structured version Visualization version GIF version | ||
| Description: A nonnegative number plus one is a positive number. (Contributed by Mario Carneiro, 28-May-2016.) |
| Ref | Expression |
|---|---|
| rpgecld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ge0p1rp.2 | ⊢ (𝜑 → 0 ≤ 𝐴) |
| Ref | Expression |
|---|---|
| ge0p1rpd | ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rpgecld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 2 | ge0p1rp.2 | . 2 ⊢ (𝜑 → 0 ≤ 𝐴) | |
| 3 | ge0p1rp 13066 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 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 ax-pre-mulgt0 11232 |
| 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-reu 3381 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-rp 13035 |
| This theorem is referenced by: lo1bddrp 15561 o1rlimmul 15655 mertenslem1 15920 mertenslem2 15921 nlmvscnlem2 24706 nlmvscnlem1 24707 nghmcn 24766 cnheibor 24987 ipcnlem2 25278 ipcnlem1 25279 pjthlem1 25471 itg2const2 25776 itgulm 26451 abelthlem8 26483 loglesqrt 26804 logdiflbnd 27038 ftalem4 27119 logfacrlim 27268 dchrisumlem3 27535 pntrsumo1 27609 smcnlem 30716 pjhthlem1 31410 faclimlem1 35743 faclimlem3 35745 faclim 35746 iprodfac 35747 isbnd3 37791 totbndbnd 37796 rrntotbnd 37843 aks4d1p1p7 42075 wallispilem4 46083 wallispi 46085 wallispi2lem1 46086 stirlinglem1 46089 stirlinglem4 46092 stirlinglem6 46094 stirlinglem10 46098 stirlinglem11 46099 stirlinglem12 46100 stirlinglem13 46101 fourierdlem30 46152 fourierdlem77 46198 |
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