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Mirrors > Home > MPE Home > Th. List > modmuladdim | Structured version Visualization version GIF version |
Description: Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
Ref | Expression |
---|---|
modmuladdim | ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11988 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | modelico 13252 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (𝐴 mod 𝑀) ∈ (0[,)𝑀)) | |
3 | 1, 2 | sylan 582 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (𝐴 mod 𝑀) ∈ (0[,)𝑀)) |
4 | 3 | adantr 483 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 mod 𝑀) = 𝐵) → (𝐴 mod 𝑀) ∈ (0[,)𝑀)) |
5 | eleq1 2902 | . . . . 5 ⊢ ((𝐴 mod 𝑀) = 𝐵 → ((𝐴 mod 𝑀) ∈ (0[,)𝑀) ↔ 𝐵 ∈ (0[,)𝑀))) | |
6 | 5 | adantl 484 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 mod 𝑀) = 𝐵) → ((𝐴 mod 𝑀) ∈ (0[,)𝑀) ↔ 𝐵 ∈ (0[,)𝑀))) |
7 | 4, 6 | mpbid 234 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 mod 𝑀) = 𝐵) → 𝐵 ∈ (0[,)𝑀)) |
8 | simpll 765 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ 𝐵 ∈ (0[,)𝑀)) → 𝐴 ∈ ℤ) | |
9 | simpr 487 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ 𝐵 ∈ (0[,)𝑀)) → 𝐵 ∈ (0[,)𝑀)) | |
10 | simpr 487 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → 𝑀 ∈ ℝ+) | |
11 | 10 | adantr 483 | . . . . . 6 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ 𝐵 ∈ (0[,)𝑀)) → 𝑀 ∈ ℝ+) |
12 | modmuladd 13284 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (0[,)𝑀) ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | |
13 | 8, 9, 11, 12 | syl3anc 1367 | . . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ 𝐵 ∈ (0[,)𝑀)) → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
14 | 13 | biimpd 231 | . . . 4 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ 𝐵 ∈ (0[,)𝑀)) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
15 | 14 | impancom 454 | . . 3 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 mod 𝑀) = 𝐵) → (𝐵 ∈ (0[,)𝑀) → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
16 | 7, 15 | mpd 15 | . 2 ⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) ∧ (𝐴 mod 𝑀) = 𝐵) → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵)) |
17 | 16 | ex 415 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 ℤcz 11984 ℝ+crp 12392 [,)cico 12743 mod cmo 13240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fl 13165 df-mod 13241 |
This theorem is referenced by: modmuladdnn0 13286 2lgsoddprmlem2 25987 fppr2odd 43903 |
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