![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nn0addge1 | 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 9216 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | nn0ge0 9232 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
3 | 1, 2 | jca 306 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) |
4 | addge01 8460 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 ≤ 𝑁 ↔ 𝐴 ≤ (𝐴 + 𝑁))) | |
5 | 4 | biimp3a 1356 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) → 𝐴 ≤ (𝐴 + 𝑁)) |
6 | 5 | 3expb 1206 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) → 𝐴 ≤ (𝐴 + 𝑁)) |
7 | 3, 6 | sylan2 286 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5897 ℝcr 7841 0cc0 7842 + caddc 7845 ≤ cle 8024 ℕ0cn0 9207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-iota 5196 df-fv 5243 df-ov 5900 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-inn 8951 df-n0 9208 |
This theorem is referenced by: nn0addge1i 9255 fzctr 10165 nn0opthlem2d 10736 pcaddlem 12374 4sqexercise2 12434 4sqlemsdc 12435 |
Copyright terms: Public domain | W3C validator |