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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 12270 | . 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 2081 class class class wbr 4962 (class class class)co 7016 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 ≤ cle 10522 ℝ+crp 12239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-rp 12240 |
This theorem is referenced by: lo1bddrp 14716 o1rlimmul 14809 mertenslem1 15073 mertenslem2 15074 nlmvscnlem2 22977 nlmvscnlem1 22978 nghmcn 23037 cnheibor 23242 ipcnlem2 23530 ipcnlem1 23531 pjthlem1 23723 itg2const2 24025 itgulm 24679 abelthlem8 24710 loglesqrt 25020 logdiflbnd 25254 ftalem4 25335 logfacrlim 25482 dchrisumlem3 25749 pntrsumo1 25823 smcnlem 28165 pjhthlem1 28859 faclimlem1 32583 faclimlem3 32585 faclim 32586 iprodfac 32587 isbnd3 34594 totbndbnd 34599 rrntotbnd 34646 wallispilem4 41895 wallispi 41897 wallispi2lem1 41898 stirlinglem1 41901 stirlinglem4 41904 stirlinglem6 41906 stirlinglem10 41910 stirlinglem11 41911 stirlinglem12 41912 stirlinglem13 41913 fourierdlem30 41964 fourierdlem77 42010 |
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