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Mirrors > Home > ILE Home > Th. List > modqmul12d | GIF version |
Description: Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Jim Kingdon, 24-Oct-2021.) |
Ref | Expression |
---|---|
modqmul12d.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
modqmul12d.2 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
modqmul12d.3 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
modqmul12d.4 | ⊢ (𝜑 → 𝐷 ∈ ℤ) |
modqmul12d.5 | ⊢ (𝜑 → 𝐸 ∈ ℚ) |
modqmul12d.egt0 | ⊢ (𝜑 → 0 < 𝐸) |
modqmul12d.6 | ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) |
modqmul12d.7 | ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) |
Ref | Expression |
---|---|
modqmul12d | ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqmul12d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | zq 9210 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ) |
4 | modqmul12d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
5 | zq 9210 | . . . 4 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℚ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℚ) |
7 | modqmul12d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
8 | modqmul12d.5 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℚ) | |
9 | modqmul12d.egt0 | . . 3 ⊢ (𝜑 → 0 < 𝐸) | |
10 | modqmul12d.6 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) | |
11 | 3, 6, 7, 8, 9, 10 | modqmul1 9933 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) |
12 | 4 | zcnd 8968 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 7 | zcnd 8968 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | 12, 13 | mulcomd 7606 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
15 | 14 | oveq1d 5705 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐶 · 𝐵) mod 𝐸)) |
16 | zq 9210 | . . . . 5 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℚ) | |
17 | 7, 16 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℚ) |
18 | modqmul12d.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
19 | zq 9210 | . . . . 5 ⊢ (𝐷 ∈ ℤ → 𝐷 ∈ ℚ) | |
20 | 18, 19 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℚ) |
21 | modqmul12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
22 | 17, 20, 4, 8, 9, 21 | modqmul1 9933 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) |
23 | 18 | zcnd 8968 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
24 | 23, 12 | mulcomd 7606 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
25 | 24 | oveq1d 5705 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
26 | 15, 22, 25 | 3eqtrd 2131 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
27 | 11, 26 | eqtrd 2127 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 0cc0 7447 · cmul 7452 < clt 7619 ℤcz 8848 ℚcq 9203 mod cmo 9878 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-n0 8772 df-z 8849 df-q 9204 df-rp 9234 df-fl 9826 df-mod 9879 |
This theorem is referenced by: (None) |
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