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| Mirrors > Home > MPE Home > Th. List > Mathboxes > remexz | Structured version Visualization version GIF version | ||
| Description: Division with rest. (Contributed by metakunt, 15-May-2025.) |
| Ref | Expression |
|---|---|
| remexz.1 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| remexz.2 | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
| Ref | Expression |
|---|---|
| remexz | ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐴 − 1))𝑁 = ((𝑥 · 𝐴) + 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | remexz.1 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
| 2 | remexz.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
| 3 | zmodfzo 13934 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (𝑁 mod 𝐴) ∈ (0..^𝐴)) | |
| 4 | 1, 2, 3 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁 mod 𝐴) ∈ (0..^𝐴)) |
| 5 | 2 | nnzd 12640 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
| 6 | fzoval 13700 | . . . . 5 ⊢ (𝐴 ∈ ℤ → (0..^𝐴) = (0...(𝐴 − 1))) | |
| 7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (0..^𝐴) = (0...(𝐴 − 1))) |
| 8 | 4, 7 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → (𝑁 mod 𝐴) ∈ (0...(𝐴 − 1))) |
| 9 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 mod 𝐴)) → 𝑦 = (𝑁 mod 𝐴)) | |
| 10 | 9 | oveq2d 7447 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 mod 𝐴)) → ((𝑥 · 𝐴) + 𝑦) = ((𝑥 · 𝐴) + (𝑁 mod 𝐴))) |
| 11 | 10 | eqeq2d 2748 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 mod 𝐴)) → (𝑁 = ((𝑥 · 𝐴) + 𝑦) ↔ 𝑁 = ((𝑥 · 𝐴) + (𝑁 mod 𝐴)))) |
| 12 | 11 | rexbidv 3179 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = (𝑁 mod 𝐴)) → (∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + 𝑦) ↔ ∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + (𝑁 mod 𝐴)))) |
| 13 | eqidd 2738 | . . . 4 ⊢ (𝜑 → (𝑁 mod 𝐴) = (𝑁 mod 𝐴)) | |
| 14 | 2 | nnrpd 13075 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
| 15 | modmuladdim 13955 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℝ+) → ((𝑁 mod 𝐴) = (𝑁 mod 𝐴) → ∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + (𝑁 mod 𝐴)))) | |
| 16 | 1, 14, 15 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝑁 mod 𝐴) = (𝑁 mod 𝐴) → ∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + (𝑁 mod 𝐴)))) |
| 17 | 13, 16 | mpd 15 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + (𝑁 mod 𝐴))) |
| 18 | 8, 12, 17 | rspcedvd 3624 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ (0...(𝐴 − 1))∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + 𝑦)) |
| 19 | rexcom 3290 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐴 − 1))𝑁 = ((𝑥 · 𝐴) + 𝑦) ↔ ∃𝑦 ∈ (0...(𝐴 − 1))∃𝑥 ∈ ℤ 𝑁 = ((𝑥 · 𝐴) + 𝑦)) | |
| 20 | 18, 19 | sylibr 234 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐴 − 1))𝑁 = ((𝑥 · 𝐴) + 𝑦)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 ℕcn 12266 ℤcz 12613 ℝ+crp 13034 ...cfz 13547 ..^cfzo 13694 mod cmo 13909 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 |
| This theorem is referenced by: primrootspoweq0 42107 aks6d1c6lem5 42178 |
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