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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 9655 | . . . 4 ⊢ (𝐶 ∈ ℚ → -𝐶 ∈ ℚ) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ (𝜑 → -𝐶 ∈ ℚ) |
6 | modqadd12d.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℚ) | |
7 | qnegcl 9655 | . . . 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 10398 | . . 3 ⊢ (𝜑 → (-𝐶 mod 𝐸) = (-𝐷 mod 𝐸)) |
14 | 1, 2, 5, 8, 9, 10, 11, 13 | modqadd12d 10399 | . 2 ⊢ (𝜑 → ((𝐴 + -𝐶) mod 𝐸) = ((𝐵 + -𝐷) mod 𝐸)) |
15 | qcn 9653 | . . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
16 | 1, 15 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
17 | qcn 9653 | . . . . 5 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ) | |
18 | 3, 17 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
19 | 16, 18 | negsubd 8293 | . . 3 ⊢ (𝜑 → (𝐴 + -𝐶) = (𝐴 − 𝐶)) |
20 | 19 | oveq1d 5906 | . 2 ⊢ (𝜑 → ((𝐴 + -𝐶) mod 𝐸) = ((𝐴 − 𝐶) mod 𝐸)) |
21 | qcn 9653 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
22 | 2, 21 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
23 | qcn 9653 | . . . . 5 ⊢ (𝐷 ∈ ℚ → 𝐷 ∈ ℂ) | |
24 | 6, 23 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
25 | 22, 24 | negsubd 8293 | . . 3 ⊢ (𝜑 → (𝐵 + -𝐷) = (𝐵 − 𝐷)) |
26 | 25 | oveq1d 5906 | . 2 ⊢ (𝜑 → ((𝐵 + -𝐷) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
27 | 14, 20, 26 | 3eqtr3d 2230 | 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 4018 (class class class)co 5891 ℂcc 7828 0cc0 7830 + caddc 7833 < clt 8011 − cmin 8147 -cneg 8148 ℚcq 9638 mod cmo 10341 |
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 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 |
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 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-n0 9196 df-z 9273 df-q 9639 df-rp 9673 df-fl 10289 df-mod 10342 |
This theorem is referenced by: modqsubmod 10401 modqsubmodmod 10402 |
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