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Mirrors > Home > MPE Home > Th. List > Mathboxes > modn0mul | Structured version Visualization version GIF version |
Description: If an integer is not 0 modulo a positive integer, this integer must be the sum of the product of another integer and the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.) |
Ref | Expression |
---|---|
modn0mul | ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) ≠ 0 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11979 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ) |
3 | nnre 11639 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
4 | 3 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ) |
5 | nnne0 11665 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
6 | 5 | adantl 484 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0) |
7 | 2, 4, 6 | redivcld 11462 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) |
8 | 7 | flcld 13162 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (⌊‘(𝐴 / 𝑁)) ∈ ℤ) |
9 | 8 | adantr 483 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (⌊‘(𝐴 / 𝑁)) ∈ ℤ) |
10 | zmodfzo 13256 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 mod 𝑁) ∈ (0..^𝑁)) | |
11 | 10 | anim1i 616 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → ((𝐴 mod 𝑁) ∈ (0..^𝑁) ∧ (𝐴 mod 𝑁) ≠ 0)) |
12 | fzo1fzo0n0 13082 | . . . 4 ⊢ ((𝐴 mod 𝑁) ∈ (1..^𝑁) ↔ ((𝐴 mod 𝑁) ∈ (0..^𝑁) ∧ (𝐴 mod 𝑁) ≠ 0)) | |
13 | 11, 12 | sylibr 236 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (𝐴 mod 𝑁) ∈ (1..^𝑁)) |
14 | nnrp 12394 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
15 | 1, 14 | anim12i 614 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+)) |
16 | 15 | adantr 483 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+)) |
17 | flpmodeq 13236 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁)) = 𝐴) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁)) = 𝐴) |
19 | 18 | eqcomd 2827 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁))) |
20 | oveq1 7157 | . . . . . 6 ⊢ (𝑥 = (⌊‘(𝐴 / 𝑁)) → (𝑥 · 𝑁) = ((⌊‘(𝐴 / 𝑁)) · 𝑁)) | |
21 | 20 | oveq1d 7165 | . . . . 5 ⊢ (𝑥 = (⌊‘(𝐴 / 𝑁)) → ((𝑥 · 𝑁) + 𝑦) = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦)) |
22 | 21 | eqeq2d 2832 | . . . 4 ⊢ (𝑥 = (⌊‘(𝐴 / 𝑁)) → (𝐴 = ((𝑥 · 𝑁) + 𝑦) ↔ 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦))) |
23 | oveq2 7158 | . . . . 5 ⊢ (𝑦 = (𝐴 mod 𝑁) → (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦) = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁))) | |
24 | 23 | eqeq2d 2832 | . . . 4 ⊢ (𝑦 = (𝐴 mod 𝑁) → (𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦) ↔ 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁)))) |
25 | 22, 24 | rspc2ev 3634 | . . 3 ⊢ (((⌊‘(𝐴 / 𝑁)) ∈ ℤ ∧ (𝐴 mod 𝑁) ∈ (1..^𝑁) ∧ 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦)) |
26 | 9, 13, 19, 25 | syl3anc 1367 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦)) |
27 | 26 | ex 415 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) ≠ 0 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 / cdiv 11291 ℕcn 11632 ℤcz 11975 ℝ+crp 12383 ..^cfzo 13027 ⌊cfl 13154 mod cmo 13231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 |
This theorem is referenced by: m1modmmod 44575 |
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