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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 11258 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ) |
3 | nnre 10904 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
4 | 3 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℝ) |
5 | nnne0 10930 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) | |
6 | 5 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ≠ 0) |
7 | 2, 4, 6 | redivcld 10732 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 / 𝑁) ∈ ℝ) |
8 | 7 | flcld 12461 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (⌊‘(𝐴 / 𝑁)) ∈ ℤ) |
9 | 8 | adantr 480 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (⌊‘(𝐴 / 𝑁)) ∈ ℤ) |
10 | zmodfzo 12555 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 mod 𝑁) ∈ (0..^𝑁)) | |
11 | 10 | anim1i 590 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → ((𝐴 mod 𝑁) ∈ (0..^𝑁) ∧ (𝐴 mod 𝑁) ≠ 0)) |
12 | fzo1fzo0n0 12386 | . . . 4 ⊢ ((𝐴 mod 𝑁) ∈ (1..^𝑁) ↔ ((𝐴 mod 𝑁) ∈ (0..^𝑁) ∧ (𝐴 mod 𝑁) ≠ 0)) | |
13 | 11, 12 | sylibr 223 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (𝐴 mod 𝑁) ∈ (1..^𝑁)) |
14 | nnrp 11718 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
15 | 1, 14 | anim12i 588 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+)) |
16 | 15 | adantr 480 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → (𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+)) |
17 | flpmodeq 12535 | . . . . 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 2616 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁))) |
20 | oveq1 6556 | . . . . . 6 ⊢ (𝑥 = (⌊‘(𝐴 / 𝑁)) → (𝑥 · 𝑁) = ((⌊‘(𝐴 / 𝑁)) · 𝑁)) | |
21 | 20 | oveq1d 6564 | . . . . 5 ⊢ (𝑥 = (⌊‘(𝐴 / 𝑁)) → ((𝑥 · 𝑁) + 𝑦) = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦)) |
22 | 21 | eqeq2d 2620 | . . . 4 ⊢ (𝑥 = (⌊‘(𝐴 / 𝑁)) → (𝐴 = ((𝑥 · 𝑁) + 𝑦) ↔ 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦))) |
23 | oveq2 6557 | . . . . 5 ⊢ (𝑦 = (𝐴 mod 𝑁) → (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦) = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁))) | |
24 | 23 | eqeq2d 2620 | . . . 4 ⊢ (𝑦 = (𝐴 mod 𝑁) → (𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + 𝑦) ↔ 𝐴 = (((⌊‘(𝐴 / 𝑁)) · 𝑁) + (𝐴 mod 𝑁)))) |
25 | 22, 24 | rspc2ev 3295 | . . 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 1318 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 mod 𝑁) ≠ 0) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦)) |
27 | 26 | ex 449 | 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 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 / cdiv 10563 ℕcn 10897 ℤcz 11254 ℝ+crp 11708 ..^cfzo 12334 ⌊cfl 12453 mod cmo 12530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 |
This theorem is referenced by: m1modmmod 42110 |
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