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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 13002 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 + 1) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ≤ cle 11246 ℝ+crp 12971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-rp 12972 |
This theorem is referenced by: lo1bddrp 15466 o1rlimmul 15560 mertenslem1 15827 mertenslem2 15828 nlmvscnlem2 24524 nlmvscnlem1 24525 nghmcn 24584 cnheibor 24803 ipcnlem2 25094 ipcnlem1 25095 pjthlem1 25287 itg2const2 25593 itgulm 26261 abelthlem8 26293 loglesqrt 26609 logdiflbnd 26843 ftalem4 26924 logfacrlim 27073 dchrisumlem3 27340 pntrsumo1 27414 smcnlem 30419 pjhthlem1 31113 faclimlem1 35208 faclimlem3 35210 faclim 35211 iprodfac 35212 isbnd3 37142 totbndbnd 37147 rrntotbnd 37194 aks4d1p1p7 41432 wallispilem4 45269 wallispi 45271 wallispi2lem1 45272 stirlinglem1 45275 stirlinglem4 45278 stirlinglem6 45280 stirlinglem10 45284 stirlinglem11 45285 stirlinglem12 45286 stirlinglem13 45287 fourierdlem30 45338 fourierdlem77 45384 |
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