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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 12673 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | modmul12d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
4 | 3 | zred 12673 | . . 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 13896 | . . 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 1380 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) |
10 | 3 | zcnd 12674 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
11 | 5 | zcnd 12674 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 10, 11 | mulcomd 11242 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
13 | 12 | oveq1d 7427 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐶 · 𝐵) mod 𝐸)) |
14 | 5 | zred 12673 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
15 | modmul12d.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
16 | 15 | zred 12673 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
17 | modmul12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
18 | modmul1 13896 | . . . 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 1380 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) |
20 | 15 | zcnd 12674 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
21 | 20, 10 | mulcomd 11242 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
22 | 21 | oveq1d 7427 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
23 | 13, 19, 22 | 3eqtrd 2775 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
24 | 9, 23 | eqtrd 2771 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 (class class class)co 7412 ℝcr 11115 · cmul 11121 ℤcz 12565 ℝ+crp 12981 mod cmo 13841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fl 13764 df-mod 13842 |
This theorem is referenced by: modexp 14208 fprodmodd 15948 smumul 16441 modxai 17008 elqaalem2 26173 lgsdir2lem5 27177 lgseisenlem2 27224 lgseisenlem3 27225 modexp2m1d 46742 |
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