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Mirrors > Home > MPE Home > Th. List > modadd12d | Structured version Visualization version GIF version |
Description: Additive 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 |
---|---|
modadd12d | ⊢ (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 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | modadd12d.5 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℝ+) | |
5 | modadd12d.6 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) | |
6 | modadd1 13264 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐸 ∈ ℝ+) ∧ (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐶) mod 𝐸)) | |
7 | 1, 2, 3, 4, 5, 6 | syl221anc 1373 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐶) mod 𝐸)) |
8 | 2 | recnd 10657 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 3 | recnd 10657 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
10 | 8, 9 | addcomd 10830 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐵)) |
11 | 10 | oveq1d 7160 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐶) mod 𝐸) = ((𝐶 + 𝐵) mod 𝐸)) |
12 | modadd12d.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
13 | modadd12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
14 | modadd1 13264 | . . . 4 ⊢ (((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ (𝐵 ∈ ℝ ∧ 𝐸 ∈ ℝ+) ∧ (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) → ((𝐶 + 𝐵) mod 𝐸) = ((𝐷 + 𝐵) mod 𝐸)) | |
15 | 3, 12, 2, 4, 13, 14 | syl221anc 1373 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐵) mod 𝐸) = ((𝐷 + 𝐵) mod 𝐸)) |
16 | 12 | recnd 10657 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
17 | 16, 8 | addcomd 10830 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝐵) = (𝐵 + 𝐷)) |
18 | 17 | oveq1d 7160 | . . 3 ⊢ (𝜑 → ((𝐷 + 𝐵) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
19 | 11, 15, 18 | 3eqtrd 2857 | . 2 ⊢ (𝜑 → ((𝐵 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
20 | 7, 19 | eqtrd 2853 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℝcr 10524 + caddc 10528 ℝ+crp 12377 mod cmo 13225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 |
This theorem is referenced by: modsub12d 13284 sadasslem 15807 |
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