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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 11635 | . . . 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 11922 | . . . . . 6 ⊢ (𝑀 + 𝐵) ∈ ℕ0 |
7 | 6 | nn0rei 11896 | . . . . 5 ⊢ (𝑀 + 𝐵) ∈ ℝ |
8 | 3, 7 | eqeltrri 2907 | . . . 4 ⊢ 𝐾 ∈ ℝ |
9 | 2, 8 | pm3.2i 471 | . . 3 ⊢ (𝐴 ∈ ℝ ∧ 𝐾 ∈ ℝ) |
10 | 5 | nn0rei 11896 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
11 | 10 | renegcli 10935 | . . . 4 ⊢ -𝐵 ∈ ℝ |
12 | modsubi.1 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
13 | nnrp 12388 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ 𝑁 ∈ ℝ+ |
15 | 11, 14 | pm3.2i 471 | . . 3 ⊢ (-𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ+) |
16 | modsubi.6 | . . 3 ⊢ (𝐴 mod 𝑁) = (𝐾 mod 𝑁) | |
17 | modadd1 13264 | . . 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 1452 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐾 + -𝐵) mod 𝑁) |
19 | 1 | nncni 11636 | . . . 4 ⊢ 𝐴 ∈ ℂ |
20 | 5 | nn0cni 11897 | . . . 4 ⊢ 𝐵 ∈ ℂ |
21 | 19, 20 | negsubi 10952 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
22 | 21 | oveq1i 7155 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐴 − 𝐵) mod 𝑁) |
23 | 8 | recni 10643 | . . . . 5 ⊢ 𝐾 ∈ ℂ |
24 | 23, 20 | negsubi 10952 | . . . 4 ⊢ (𝐾 + -𝐵) = (𝐾 − 𝐵) |
25 | 4 | nn0cni 11897 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
26 | 23, 20, 25 | subadd2i 10962 | . . . . 5 ⊢ ((𝐾 − 𝐵) = 𝑀 ↔ (𝑀 + 𝐵) = 𝐾) |
27 | 3, 26 | mpbir 232 | . . . 4 ⊢ (𝐾 − 𝐵) = 𝑀 |
28 | 24, 27 | eqtri 2841 | . . 3 ⊢ (𝐾 + -𝐵) = 𝑀 |
29 | 28 | oveq1i 7155 | . 2 ⊢ ((𝐾 + -𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
30 | 18, 22, 29 | 3eqtr3i 2849 | 1 ⊢ ((𝐴 − 𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 + caddc 10528 − cmin 10858 -cneg 10859 ℕcn 11626 ℕ0cn0 11885 ℝ+crp 12377 mod cmo 13225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 |
This theorem is referenced by: 1259lem5 16456 2503lem3 16460 4001lem4 16465 |
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