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Mirrors > Home > MPE Home > Th. List > modmul12d | Structured version Visualization version GIF version |
Description: Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 5-Feb-2015.) |
Ref | Expression |
---|---|
modmul12d.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
modmul12d.2 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
modmul12d.3 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
modmul12d.4 | ⊢ (𝜑 → 𝐷 ∈ ℤ) |
modmul12d.5 | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
modmul12d.6 | ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) |
modmul12d.7 | ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) |
Ref | Expression |
---|---|
modmul12d | ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modmul12d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 1 | zred 11482 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | modmul12d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
4 | 3 | zred 11482 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | modmul12d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
6 | modmul12d.5 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
7 | modmul12d.6 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) | |
8 | modmul1 12723 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℤ ∧ 𝐸 ∈ ℝ+) ∧ (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) | |
9 | 2, 4, 5, 6, 7, 8 | syl221anc 1337 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) |
10 | 3 | zcnd 11483 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | 5 | zcnd 11483 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 10, 11 | mulcomd 10061 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
13 | 12 | oveq1d 6665 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐶 · 𝐵) mod 𝐸)) |
14 | 5 | zred 11482 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
15 | modmul12d.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
16 | 15 | zred 11482 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
17 | modmul12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
18 | modmul1 12723 | . . . 4 ⊢ (((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (𝐵 ∈ ℤ ∧ 𝐸 ∈ ℝ+) ∧ (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) | |
19 | 14, 16, 3, 6, 17, 18 | syl221anc 1337 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) |
20 | 15 | zcnd 11483 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
21 | 20, 10 | mulcomd 10061 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
22 | 21 | oveq1d 6665 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
23 | 13, 19, 22 | 3eqtrd 2660 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
24 | 9, 23 | eqtrd 2656 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℝcr 9935 · cmul 9941 ℤcz 11377 ℝ+crp 11832 mod cmo 12668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-mod 12669 |
This theorem is referenced by: modexp 12999 fprodmodd 14728 smumul 15215 modxai 15772 elqaalem2 24075 lgsdir2lem5 25054 lgseisenlem2 25101 lgseisenlem3 25102 modexp2m1d 41529 |
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