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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 13270 | . . 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 1377 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐶) mod 𝐸)) |
8 | 2 | recnd 10663 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
9 | 3 | recnd 10663 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
10 | 8, 9 | addcomd 10836 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐵)) |
11 | 10 | oveq1d 7165 | . . 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 13270 | . . . 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 1377 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐵) mod 𝐸) = ((𝐷 + 𝐵) mod 𝐸)) |
16 | 12 | recnd 10663 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
17 | 16, 8 | addcomd 10836 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝐵) = (𝐵 + 𝐷)) |
18 | 17 | oveq1d 7165 | . . 3 ⊢ (𝜑 → ((𝐷 + 𝐵) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
19 | 11, 15, 18 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → ((𝐵 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
20 | 7, 19 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℝcr 10530 + caddc 10534 ℝ+crp 12383 mod cmo 13231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13156 df-mod 13232 |
This theorem is referenced by: modsub12d 13290 sadasslem 15813 |
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