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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 9004 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ) |
4 | modqmul12d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
5 | zq 9004 | . . . 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 9671 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) |
12 | 4 | zcnd 8763 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 7 | zcnd 8763 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | 12, 13 | mulcomd 7410 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
15 | 14 | oveq1d 5604 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐶 · 𝐵) mod 𝐸)) |
16 | zq 9004 | . . . . 5 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℚ) | |
17 | 7, 16 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℚ) |
18 | modqmul12d.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
19 | zq 9004 | . . . . 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 9671 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) |
23 | 18 | zcnd 8763 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
24 | 23, 12 | mulcomd 7410 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
25 | 24 | oveq1d 5604 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
26 | 15, 22, 25 | 3eqtrd 2119 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
27 | 11, 26 | eqtrd 2115 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5589 0cc0 7251 · cmul 7256 < clt 7423 ℤcz 8644 ℚcq 8997 mod cmo 9616 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 ax-arch 7365 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-po 4086 df-iso 4087 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-n0 8564 df-z 8645 df-q 8998 df-rp 9028 df-fl 9564 df-mod 9617 |
This theorem is referenced by: (None) |
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