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Mirrors > Home > MPE Home > Th. List > nn0ledivnn | Structured version Visualization version GIF version |
Description: Division of a nonnegative integer by a positive integer is less than or equal to the integer. (Contributed by AV, 19-Jun-2021.) |
Ref | Expression |
---|---|
nn0ledivnn | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12416 | . . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
2 | nnge1 12182 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 1 ≤ 𝐵) | |
3 | 2 | adantl 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → 1 ≤ 𝐵) |
4 | nnrp 12927 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ+) | |
5 | nnledivrp 13028 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℝ+) → (1 ≤ 𝐵 ↔ (𝐴 / 𝐵) ≤ 𝐴)) | |
6 | 4, 5 | sylan2 594 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (1 ≤ 𝐵 ↔ (𝐴 / 𝐵) ≤ 𝐴)) |
7 | 3, 6 | mpbid 231 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴) |
8 | 7 | ex 414 | . . . 4 ⊢ (𝐴 ∈ ℕ → (𝐵 ∈ ℕ → (𝐴 / 𝐵) ≤ 𝐴)) |
9 | nncn 12162 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ) | |
10 | nnne0 12188 | . . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
11 | 9, 10 | jca 513 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
12 | 11 | adantl 483 | . . . . . . . 8 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
13 | div0 11844 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (0 / 𝐵) = 0) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (0 / 𝐵) = 0) |
15 | 0le0 12255 | . . . . . . 7 ⊢ 0 ≤ 0 | |
16 | 14, 15 | eqbrtrdi 5145 | . . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (0 / 𝐵) ≤ 0) |
17 | oveq1 7365 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 / 𝐵) = (0 / 𝐵)) | |
18 | id 22 | . . . . . . . 8 ⊢ (𝐴 = 0 → 𝐴 = 0) | |
19 | 17, 18 | breq12d 5119 | . . . . . . 7 ⊢ (𝐴 = 0 → ((𝐴 / 𝐵) ≤ 𝐴 ↔ (0 / 𝐵) ≤ 0)) |
20 | 19 | adantr 482 | . . . . . 6 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → ((𝐴 / 𝐵) ≤ 𝐴 ↔ (0 / 𝐵) ≤ 0)) |
21 | 16, 20 | mpbird 257 | . . . . 5 ⊢ ((𝐴 = 0 ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴) |
22 | 21 | ex 414 | . . . 4 ⊢ (𝐴 = 0 → (𝐵 ∈ ℕ → (𝐴 / 𝐵) ≤ 𝐴)) |
23 | 8, 22 | jaoi 856 | . . 3 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐵 ∈ ℕ → (𝐴 / 𝐵) ≤ 𝐴)) |
24 | 1, 23 | sylbi 216 | . 2 ⊢ (𝐴 ∈ ℕ0 → (𝐵 ∈ ℕ → (𝐴 / 𝐵) ≤ 𝐴)) |
25 | 24 | imp 408 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (𝐴 / 𝐵) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5106 (class class class)co 7358 ℂcc 11050 0cc0 11052 1c1 11053 ≤ cle 11191 / cdiv 11813 ℕcn 12154 ℕ0cn0 12414 ℝ+crp 12916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-n0 12415 df-rp 12917 |
This theorem is referenced by: 2lgslem1c 26744 |
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