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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 11358 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | modmul12d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
4 | 3 | zred 11358 | . . 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 12585 | . . 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 1329 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) |
10 | 3 | zcnd 11359 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | 5 | zcnd 11359 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 10, 11 | mulcomd 9940 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
13 | 12 | oveq1d 6564 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐶 · 𝐵) mod 𝐸)) |
14 | 5 | zred 11358 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
15 | modmul12d.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
16 | 15 | zred 11358 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
17 | modmul12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
18 | modmul1 12585 | . . . 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 1329 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) |
20 | 15 | zcnd 11359 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
21 | 20, 10 | mulcomd 9940 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
22 | 21 | oveq1d 6564 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
23 | 13, 19, 22 | 3eqtrd 2648 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
24 | 9, 23 | eqtrd 2644 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℝcr 9814 · cmul 9820 ℤcz 11254 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: modexp 12861 fprodmodd 14567 smumul 15053 modxai 15610 elqaalem2 23879 lgsdir2lem5 24854 lgseisenlem2 24901 lgseisenlem3 24902 modexp2m1d 40067 |
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