Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > modqadd12d | GIF version |
Description: Additive 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 |
---|---|
modqadd12d | ⊢ (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 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℚ) | |
4 | modqadd12d.5 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ℚ) | |
5 | modqadd12d.egt0 | . . 3 ⊢ (𝜑 → 0 < 𝐸) | |
6 | modqadd12d.6 | . . 3 ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) | |
7 | 1, 2, 3, 4, 5, 6 | modqadd1 10127 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐶) mod 𝐸)) |
8 | qcn 9419 | . . . . . 6 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
9 | 2, 8 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
10 | qcn 9419 | . . . . . 6 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ) | |
11 | 3, 10 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
12 | 9, 11 | addcomd 7906 | . . . 4 ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐵)) |
13 | 12 | oveq1d 5782 | . . 3 ⊢ (𝜑 → ((𝐵 + 𝐶) mod 𝐸) = ((𝐶 + 𝐵) mod 𝐸)) |
14 | modqadd12d.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℚ) | |
15 | modqadd12d.7 | . . . 4 ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) | |
16 | 3, 14, 2, 4, 5, 15 | modqadd1 10127 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐵) mod 𝐸) = ((𝐷 + 𝐵) mod 𝐸)) |
17 | qcn 9419 | . . . . . 6 ⊢ (𝐷 ∈ ℚ → 𝐷 ∈ ℂ) | |
18 | 14, 17 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
19 | 18, 9 | addcomd 7906 | . . . 4 ⊢ (𝜑 → (𝐷 + 𝐵) = (𝐵 + 𝐷)) |
20 | 19 | oveq1d 5782 | . . 3 ⊢ (𝜑 → ((𝐷 + 𝐵) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
21 | 13, 16, 20 | 3eqtrd 2174 | . 2 ⊢ (𝜑 → ((𝐵 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
22 | 7, 21 | eqtrd 2170 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 0cc0 7613 + caddc 7616 < clt 7793 ℚcq 9404 mod cmo 10088 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-n0 8971 df-z 9048 df-q 9405 df-rp 9435 df-fl 10036 df-mod 10089 |
This theorem is referenced by: modqsub12d 10147 |
Copyright terms: Public domain | W3C validator |