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Mirrors > Home > MPE Home > Th. List > modsub12d | Structured version Visualization version GIF version |
Description: Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.) |
Ref | Expression |
---|---|
modadd12d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
modadd12d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
modadd12d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
modadd12d.4 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
modadd12d.5 | ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
modadd12d.6 | ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) |
modadd12d.7 | ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) |
Ref | Expression |
---|---|
modsub12d | ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modadd12d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | modadd12d.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | modadd12d.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 3 | renegcld 11628 | . . 3 ⊢ (𝜑 → -𝐶 ∈ ℝ) |
5 | modadd12d.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
6 | 5 | renegcld 11628 | . . 3 ⊢ (𝜑 → -𝐷 ∈ ℝ) |
7 | modadd12d.5 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
8 | modadd12d.6 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) | |
9 | modadd12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
10 | 3, 5, 7, 9 | modnegd 13878 | . . 3 ⊢ (𝜑 → (-𝐶 mod 𝐸) = (-𝐷 mod 𝐸)) |
11 | 1, 2, 4, 6, 7, 8, 10 | modadd12d 13879 | . 2 ⊢ (𝜑 → ((𝐴 + -𝐶) mod 𝐸) = ((𝐵 + -𝐷) mod 𝐸)) |
12 | 1 | recnd 11229 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
13 | 3 | recnd 11229 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
14 | 12, 13 | negsubd 11564 | . . 3 ⊢ (𝜑 → (𝐴 + -𝐶) = (𝐴 − 𝐶)) |
15 | 14 | oveq1d 7411 | . 2 ⊢ (𝜑 → ((𝐴 + -𝐶) mod 𝐸) = ((𝐴 − 𝐶) mod 𝐸)) |
16 | 2 | recnd 11229 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
17 | 5 | recnd 11229 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
18 | 16, 17 | negsubd 11564 | . . 3 ⊢ (𝜑 → (𝐵 + -𝐷) = (𝐵 − 𝐷)) |
19 | 18 | oveq1d 7411 | . 2 ⊢ (𝜑 → ((𝐵 + -𝐷) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
20 | 11, 15, 19 | 3eqtr3d 2781 | 1 ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 (class class class)co 7396 ℝcr 11096 + caddc 11100 − cmin 11431 -cneg 11432 ℝ+crp 12961 mod cmo 13821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-n0 12460 df-z 12546 df-uz 12810 df-rp 12962 df-fl 13744 df-mod 13822 |
This theorem is referenced by: modsubmod 13881 modsubmodmod 13882 fermltlchr 32440 znfermltl 32441 proththd 46155 |
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