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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 9662 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ) |
4 | modqmul12d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
5 | zq 9662 | . . . 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 10414 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐶) mod 𝐸)) |
12 | 4 | zcnd 9411 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 7 | zcnd 9411 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | 12, 13 | mulcomd 8014 | . . . 4 ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
15 | 14 | oveq1d 5915 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐶 · 𝐵) mod 𝐸)) |
16 | zq 9662 | . . . . 5 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ ℚ) | |
17 | 7, 16 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℚ) |
18 | modqmul12d.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℤ) | |
19 | zq 9662 | . . . . 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 10414 | . . 3 ⊢ (𝜑 → ((𝐶 · 𝐵) mod 𝐸) = ((𝐷 · 𝐵) mod 𝐸)) |
23 | 18 | zcnd 9411 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
24 | 23, 12 | mulcomd 8014 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐵) = (𝐵 · 𝐷)) |
25 | 24 | oveq1d 5915 | . . 3 ⊢ (𝜑 → ((𝐷 · 𝐵) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
26 | 15, 22, 25 | 3eqtrd 2226 | . 2 ⊢ (𝜑 → ((𝐵 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
27 | 11, 26 | eqtrd 2222 | 1 ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 (class class class)co 5900 0cc0 7846 · cmul 7851 < clt 8027 ℤcz 9288 ℚcq 9655 mod cmo 10359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-id 4314 df-po 4317 df-iso 4318 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-n0 9212 df-z 9289 df-q 9656 df-rp 9690 df-fl 10307 df-mod 10360 |
This theorem is referenced by: modqexp 10687 fprodmodd 11690 lgsdir2lem5 14919 lgseisenlem2 14937 |
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