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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 10906 | . . . 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 11207 | . . . . . 6 ⊢ (𝑀 + 𝐵) ∈ ℕ0 |
7 | 6 | nn0rei 11180 | . . . . 5 ⊢ (𝑀 + 𝐵) ∈ ℝ |
8 | 3, 7 | eqeltrri 2685 | . . . 4 ⊢ 𝐾 ∈ ℝ |
9 | 2, 8 | pm3.2i 470 | . . 3 ⊢ (𝐴 ∈ ℝ ∧ 𝐾 ∈ ℝ) |
10 | 5 | nn0rei 11180 | . . . . 5 ⊢ 𝐵 ∈ ℝ |
11 | 10 | renegcli 10221 | . . . 4 ⊢ -𝐵 ∈ ℝ |
12 | modsubi.1 | . . . . 5 ⊢ 𝑁 ∈ ℕ | |
13 | nnrp 11718 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ+) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ 𝑁 ∈ ℝ+ |
15 | 11, 14 | pm3.2i 470 | . . 3 ⊢ (-𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ+) |
16 | modsubi.6 | . . 3 ⊢ (𝐴 mod 𝑁) = (𝐾 mod 𝑁) | |
17 | modadd1 12569 | . . 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 1416 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐾 + -𝐵) mod 𝑁) |
19 | 1 | nncni 10907 | . . . 4 ⊢ 𝐴 ∈ ℂ |
20 | 5 | nn0cni 11181 | . . . 4 ⊢ 𝐵 ∈ ℂ |
21 | 19, 20 | negsubi 10238 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
22 | 21 | oveq1i 6559 | . 2 ⊢ ((𝐴 + -𝐵) mod 𝑁) = ((𝐴 − 𝐵) mod 𝑁) |
23 | 8 | recni 9931 | . . . . 5 ⊢ 𝐾 ∈ ℂ |
24 | 23, 20 | negsubi 10238 | . . . 4 ⊢ (𝐾 + -𝐵) = (𝐾 − 𝐵) |
25 | 4 | nn0cni 11181 | . . . . . 6 ⊢ 𝑀 ∈ ℂ |
26 | 23, 20, 25 | subadd2i 10248 | . . . . 5 ⊢ ((𝐾 − 𝐵) = 𝑀 ↔ (𝑀 + 𝐵) = 𝐾) |
27 | 3, 26 | mpbir 220 | . . . 4 ⊢ (𝐾 − 𝐵) = 𝑀 |
28 | 24, 27 | eqtri 2632 | . . 3 ⊢ (𝐾 + -𝐵) = 𝑀 |
29 | 28 | oveq1i 6559 | . 2 ⊢ ((𝐾 + -𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
30 | 18, 22, 29 | 3eqtr3i 2640 | 1 ⊢ ((𝐴 − 𝐵) mod 𝑁) = (𝑀 mod 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℝcr 9814 + caddc 9818 − cmin 10145 -cneg 10146 ℕcn 10897 ℕ0cn0 11169 ℝ+crp 11708 mod cmo 12530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 |
This theorem is referenced by: 1259lem5 15680 2503lem3 15684 4001lem4 15689 |
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