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Mirrors > Home > MPE Home > Th. List > nn0addge1 | Structured version Visualization version GIF version |
Description: A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.) |
Ref | Expression |
---|---|
nn0addge1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12123 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | nn0ge0 12139 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
3 | 1, 2 | jca 515 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) |
4 | addge01 11366 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ 𝐴 ≤ (𝐴 + 𝑁))) | |
5 | 4 | biimp3a 1471 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) → 𝐴 ≤ (𝐴 + 𝑁)) |
6 | 5 | 3expb 1122 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) → 𝐴 ≤ (𝐴 + 𝑁)) |
7 | 3, 6 | sylan2 596 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5067 (class class class)co 7231 ℝcr 10752 0cc0 10753 + caddc 10756 ≤ cle 10892 ℕ0cn0 12114 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-er 8411 df-en 8647 df-dom 8648 df-sdom 8649 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-n0 12115 |
This theorem is referenced by: nn0addge1i 12162 eluzmn 12469 fzctr 13248 elincfzoext 13324 pcaddlem 16465 psgnunilem2 18911 mndodconglem 18957 efgredleme 19157 mplmonmul 21017 coe1tmmul2fv 21223 coe1pwmulfv 21225 uniioombllem3 24506 coe1mul3 25021 ply1divmo 25057 plydiveu 25215 quotcan 25226 leibpi 25849 basellem6 25992 chtublem 26116 pntrmax 26469 signstfvc 32289 prodsplit 39912 fourierdlem47 43397 |
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