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Mirrors > Home > ILE Home > Th. List > modqsub12d | GIF version |
Description: Subtraction property of the modulo operation. (Contributed by Jim Kingdon, 25-Oct-2021.) |
Ref | Expression |
---|---|
modqadd12d.1 | ⊢ (𝜑 → 𝐴 ∈ ℚ) |
modqadd12d.2 | ⊢ (𝜑 → 𝐵 ∈ ℚ) |
modqadd12d.3 | ⊢ (𝜑 → 𝐶 ∈ ℚ) |
modqadd12d.4 | ⊢ (𝜑 → 𝐷 ∈ ℚ) |
modqadd12d.5 | ⊢ (𝜑 → 𝐸 ∈ ℚ) |
modqadd12d.egt0 | ⊢ (𝜑 → 0 < 𝐸) |
modqadd12d.6 | ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) |
modqadd12d.7 | ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) |
Ref | Expression |
---|---|
modqsub12d | ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqadd12d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℚ) | |
2 | modqadd12d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℚ) | |
3 | modqadd12d.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℚ) | |
4 | qnegcl 9582 | . . . 4 ⊢ (𝐶 ∈ ℚ → -𝐶 ∈ ℚ) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ (𝜑 → -𝐶 ∈ ℚ) |
6 | modqadd12d.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℚ) | |
7 | qnegcl 9582 | . . . 4 ⊢ (𝐷 ∈ ℚ → -𝐷 ∈ ℚ) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (𝜑 → -𝐷 ∈ ℚ) |
9 | modqadd12d.5 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℚ) | |
10 | modqadd12d.egt0 | . . 3 ⊢ (𝜑 → 0 < 𝐸) | |
11 | modqadd12d.6 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) | |
12 | modqadd12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
13 | 3, 6, 9, 10, 12 | modqnegd 10322 | . . 3 ⊢ (𝜑 → (-𝐶 mod 𝐸) = (-𝐷 mod 𝐸)) |
14 | 1, 2, 5, 8, 9, 10, 11, 13 | modqadd12d 10323 | . 2 ⊢ (𝜑 → ((𝐴 + -𝐶) mod 𝐸) = ((𝐵 + -𝐷) mod 𝐸)) |
15 | qcn 9580 | . . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
16 | 1, 15 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
17 | qcn 9580 | . . . . 5 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ) | |
18 | 3, 17 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
19 | 16, 18 | negsubd 8223 | . . 3 ⊢ (𝜑 → (𝐴 + -𝐶) = (𝐴 − 𝐶)) |
20 | 19 | oveq1d 5865 | . 2 ⊢ (𝜑 → ((𝐴 + -𝐶) mod 𝐸) = ((𝐴 − 𝐶) mod 𝐸)) |
21 | qcn 9580 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
22 | 2, 21 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
23 | qcn 9580 | . . . . 5 ⊢ (𝐷 ∈ ℚ → 𝐷 ∈ ℂ) | |
24 | 6, 23 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
25 | 22, 24 | negsubd 8223 | . . 3 ⊢ (𝜑 → (𝐵 + -𝐷) = (𝐵 − 𝐷)) |
26 | 25 | oveq1d 5865 | . 2 ⊢ (𝜑 → ((𝐵 + -𝐷) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
27 | 14, 20, 26 | 3eqtr3d 2211 | 1 ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 (class class class)co 5850 ℂcc 7759 0cc0 7761 + caddc 7764 < clt 7941 − cmin 8077 -cneg 8078 ℚcq 9565 mod cmo 10265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 ax-arch 7880 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-n0 9123 df-z 9200 df-q 9566 df-rp 9598 df-fl 10213 df-mod 10266 |
This theorem is referenced by: modqsubmod 10325 modqsubmodmod 10326 |
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