![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > modsubi | Structured version Visualization version GIF version |
Description: Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
modsubi.1 | ⊢ 𝑁 ∈ ℕ |
modsubi.2 | ⊢ 𝐴 ∈ ℕ |
modsubi.3 | ⊢ 𝐵 ∈ ℕ0 |
modsubi.4 | ⊢ 𝑀 ∈ ℕ0 |
modsubi.6 | ⊢ (𝐴 mod 𝑁) = (𝐾 mod 𝑁) |
modsubi.5 | ⊢ (𝑀 + 𝐵) = 𝐾 |
Ref | Expression |
---|---|
modsubi | ⊢ ((𝐴 − 𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modsubi.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ | |
2 | 1 | nnrei 11641 | . . . 4 ⊢ 𝐴 ∈ ℝ |
3 | modsubi.5 | . . . . 5 ⊢ (𝑀 + 𝐵) = 𝐾 | |
4 | modsubi.4 | . . . . . . 7 ⊢ 𝑀 ∈ ℕ0 | |
5 | modsubi.3 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
6 | 4, 5 | nn0addcli 11928 | . . . . . 6 ⊢ (𝑀 + 𝐵) ∈ ℕ0 |
7 | 6 | nn0rei 11902 | . . . . 5 ⊢ (𝑀 + 𝐵) ∈ ℝ |
8 | 3, 7 | eqeltrri 2910 | . . . 4 ⊢ 𝐾 ∈ ℝ |
9 | 2, 8 | pm3.2i 473 | . . 3 ⊢ (𝐴 ∈ ℝ ∧ 𝐾 ∈ ℝ) |
10 | 5 | nn0rei 11902 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
11 | 10 | renegcli 10941 | . . . 4 ⊢ -𝐵 ∈ ℝ |
12 | modsubi.1 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
13 | nnrp 12394 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ 𝑁 ∈ ℝ+ |
15 | 11, 14 | pm3.2i 473 | . . 3 ⊢ (-𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ+) |
16 | modsubi.6 | . . 3 ⊢ (𝐴 mod 𝑁) = (𝐾 mod 𝑁) | |
17 | modadd1 13270 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐾 ∈ ℝ) ∧ (-𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ+) ∧ (𝐴 mod 𝑁) = (𝐾 mod 𝑁)) → ((𝐴 + -𝐵) mod 𝑁) = ((𝐾 + -𝐵) mod 𝑁)) | |
18 | 9, 15, 16, 17 | mp3an 1457 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐾 + -𝐵) mod 𝑁) |
19 | 1 | nncni 11642 | . . . 4 ⊢ 𝐴 ∈ ℂ |
20 | 5 | nn0cni 11903 | . . . 4 ⊢ 𝐵 ∈ ℂ |
21 | 19, 20 | negsubi 10958 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
22 | 21 | oveq1i 7160 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐴 − 𝐵) mod 𝑁) |
23 | 8 | recni 10649 | . . . . 5 ⊢ 𝐾 ∈ ℂ |
24 | 23, 20 | negsubi 10958 | . . . 4 ⊢ (𝐾 + -𝐵) = (𝐾 − 𝐵) |
25 | 4 | nn0cni 11903 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
26 | 23, 20, 25 | subadd2i 10968 | . . . . 5 ⊢ ((𝐾 − 𝐵) = 𝑀 ↔ (𝑀 + 𝐵) = 𝐾) |
27 | 3, 26 | mpbir 233 | . . . 4 ⊢ (𝐾 − 𝐵) = 𝑀 |
28 | 24, 27 | eqtri 2844 | . . 3 ⊢ (𝐾 + -𝐵) = 𝑀 |
29 | 28 | oveq1i 7160 | . 2 ⊢ ((𝐾 + -𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
30 | 18, 22, 29 | 3eqtr3i 2852 | 1 ⊢ ((𝐴 − 𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℝcr 10530 + caddc 10534 − cmin 10864 -cneg 10865 ℕcn 11632 ℕ0cn0 11891 ℝ+crp 12383 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-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-fl 13156 df-mod 13232 |
This theorem is referenced by: 1259lem5 16462 2503lem3 16466 4001lem4 16471 |
Copyright terms: Public domain | W3C validator |