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Mirrors > Home > ILE Home > Th. List > modqm1p1mod0 | GIF version |
Description: If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.) |
Ref | Expression |
---|---|
modqm1p1mod0 | ⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1000 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → 𝐴 ∈ ℚ) | |
2 | 1z 9277 | . . . . 5 ⊢ 1 ∈ ℤ | |
3 | zq 9624 | . . . . 5 ⊢ (1 ∈ ℤ → 1 ∈ ℚ) | |
4 | 2, 3 | mp1i 10 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → 1 ∈ ℚ) |
5 | simp2 998 | . . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → 𝑀 ∈ ℚ) | |
6 | 5 | adantr 276 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → 𝑀 ∈ ℚ) |
7 | simpl3 1002 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → 0 < 𝑀) | |
8 | modqaddmod 10360 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) ∧ (𝑀 ∈ ℚ ∧ 0 < 𝑀)) → (((𝐴 mod 𝑀) + 1) mod 𝑀) = ((𝐴 + 1) mod 𝑀)) | |
9 | 1, 4, 6, 7, 8 | syl22anc 1239 | . . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → (((𝐴 mod 𝑀) + 1) mod 𝑀) = ((𝐴 + 1) mod 𝑀)) |
10 | oveq1 5881 | . . . . . 6 ⊢ ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 mod 𝑀) + 1) = ((𝑀 − 1) + 1)) | |
11 | 10 | oveq1d 5889 | . . . . 5 ⊢ ((𝐴 mod 𝑀) = (𝑀 − 1) → (((𝐴 mod 𝑀) + 1) mod 𝑀) = (((𝑀 − 1) + 1) mod 𝑀)) |
12 | 11 | adantl 277 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → (((𝐴 mod 𝑀) + 1) mod 𝑀) = (((𝑀 − 1) + 1) mod 𝑀)) |
13 | qcn 9632 | . . . . . . . 8 ⊢ (𝑀 ∈ ℚ → 𝑀 ∈ ℂ) | |
14 | 5, 13 | syl 14 | . . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → 𝑀 ∈ ℂ) |
15 | 14 | adantr 276 | . . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → 𝑀 ∈ ℂ) |
16 | npcan1 8333 | . . . . . 6 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) | |
17 | 15, 16 | syl 14 | . . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → ((𝑀 − 1) + 1) = 𝑀) |
18 | 17 | oveq1d 5889 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → (((𝑀 − 1) + 1) mod 𝑀) = (𝑀 mod 𝑀)) |
19 | modqid0 10347 | . . . . 5 ⊢ ((𝑀 ∈ ℚ ∧ 0 < 𝑀) → (𝑀 mod 𝑀) = 0) | |
20 | 6, 7, 19 | syl2anc 411 | . . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → (𝑀 mod 𝑀) = 0) |
21 | 12, 18, 20 | 3eqtrd 2214 | . . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → (((𝐴 mod 𝑀) + 1) mod 𝑀) = 0) |
22 | 9, 21 | eqtr3d 2212 | . 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) ∧ (𝐴 mod 𝑀) = (𝑀 − 1)) → ((𝐴 + 1) mod 𝑀) = 0) |
23 | 22 | ex 115 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℚ ∧ 0 < 𝑀) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℂcc 7808 0cc0 7810 1c1 7811 + caddc 7813 < clt 7990 − cmin 8126 ℤcz 9251 ℚcq 9617 mod cmo 10319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-n0 9175 df-z 9252 df-q 9618 df-rp 9652 df-fl 10267 df-mod 10320 |
This theorem is referenced by: (None) |
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