![]() |
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 12259 | . . . 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 12547 | . . . . . 6 ⊢ (𝑀 + 𝐵) ∈ ℕ0 |
7 | 6 | nn0rei 12521 | . . . . 5 ⊢ (𝑀 + 𝐵) ∈ ℝ |
8 | 3, 7 | eqeltrri 2826 | . . . 4 ⊢ 𝐾 ∈ ℝ |
9 | 2, 8 | pm3.2i 469 | . . 3 ⊢ (𝐴 ∈ ℝ ∧ 𝐾 ∈ ℝ) |
10 | 5 | nn0rei 12521 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
11 | 10 | renegcli 11559 | . . . 4 ⊢ -𝐵 ∈ ℝ |
12 | modsubi.1 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
13 | nnrp 13025 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ 𝑁 ∈ ℝ+ |
15 | 11, 14 | pm3.2i 469 | . . 3 ⊢ (-𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ+) |
16 | modsubi.6 | . . 3 ⊢ (𝐴 mod 𝑁) = (𝐾 mod 𝑁) | |
17 | modadd1 13913 | . . 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 12260 | . . . 4 ⊢ 𝐴 ∈ ℂ |
20 | 5 | nn0cni 12522 | . . . 4 ⊢ 𝐵 ∈ ℂ |
21 | 19, 20 | negsubi 11576 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
22 | 21 | oveq1i 7436 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐴 − 𝐵) mod 𝑁) |
23 | 8 | recni 11266 | . . . . 5 ⊢ 𝐾 ∈ ℂ |
24 | 23, 20 | negsubi 11576 | . . . 4 ⊢ (𝐾 + -𝐵) = (𝐾 − 𝐵) |
25 | 4 | nn0cni 12522 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
26 | 23, 20, 25 | subadd2i 11586 | . . . . 5 ⊢ ((𝐾 − 𝐵) = 𝑀 ↔ (𝑀 + 𝐵) = 𝐾) |
27 | 3, 26 | mpbir 230 | . . . 4 ⊢ (𝐾 − 𝐵) = 𝑀 |
28 | 24, 27 | eqtri 2756 | . . 3 ⊢ (𝐾 + -𝐵) = 𝑀 |
29 | 28 | oveq1i 7436 | . 2 ⊢ ((𝐾 + -𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
30 | 18, 22, 29 | 3eqtr3i 2764 | 1 ⊢ ((𝐴 − 𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 (class class class)co 7426 ℝcr 11145 + caddc 11149 − cmin 11482 -cneg 11483 ℕcn 12250 ℕ0cn0 12510 ℝ+crp 13014 mod cmo 13874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fl 13797 df-mod 13875 |
This theorem is referenced by: 1259lem5 17111 2503lem3 17115 4001lem4 17120 |
Copyright terms: Public domain | W3C validator |