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Mirrors > Home > MPE Home > Th. List > ex-mod | Structured version Visualization version GIF version |
Description: Example for df-mod 13588. (Contributed by AV, 3-Sep-2021.) |
Ref | Expression |
---|---|
ex-mod | ⊢ ((5 mod 3) = 2 ∧ (-7 mod 2) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3p2e5 12124 | . . . . 5 ⊢ (3 + 2) = 5 | |
2 | 1 | eqcomi 2749 | . . . 4 ⊢ 5 = (3 + 2) |
3 | 2 | oveq1i 7281 | . . 3 ⊢ (5 mod 3) = ((3 + 2) mod 3) |
4 | 2nn0 12250 | . . . 4 ⊢ 2 ∈ ℕ0 | |
5 | 3nn 12052 | . . . 4 ⊢ 3 ∈ ℕ | |
6 | 2lt3 12145 | . . . 4 ⊢ 2 < 3 | |
7 | addmodid 13637 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 2 < 3) → ((3 + 2) mod 3) = 2) | |
8 | 4, 5, 6, 7 | mp3an 1460 | . . 3 ⊢ ((3 + 2) mod 3) = 2 |
9 | 3, 8 | eqtri 2768 | . 2 ⊢ (5 mod 3) = 2 |
10 | 2re 12047 | . . . . . 6 ⊢ 2 ∈ ℝ | |
11 | 2lt7 12163 | . . . . . 6 ⊢ 2 < 7 | |
12 | 10, 11 | ltneii 11088 | . . . . 5 ⊢ 2 ≠ 7 |
13 | 2nn 12046 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
14 | 1lt2 12144 | . . . . . . 7 ⊢ 1 < 2 | |
15 | eluz2b2 12660 | . . . . . . 7 ⊢ (2 ∈ (ℤ≥‘2) ↔ (2 ∈ ℕ ∧ 1 < 2)) | |
16 | 13, 14, 15 | mpbir2an 708 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
17 | 7prm 16810 | . . . . . 6 ⊢ 7 ∈ ℙ | |
18 | dvdsprm 16406 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 7 ∈ ℙ) → (2 ∥ 7 ↔ 2 = 7)) | |
19 | 16, 17, 18 | mp2an 689 | . . . . 5 ⊢ (2 ∥ 7 ↔ 2 = 7) |
20 | 12, 19 | nemtbir 3042 | . . . 4 ⊢ ¬ 2 ∥ 7 |
21 | 2z 12352 | . . . . 5 ⊢ 2 ∈ ℤ | |
22 | 7nn 12065 | . . . . . 6 ⊢ 7 ∈ ℕ | |
23 | 22 | nnzi 12344 | . . . . 5 ⊢ 7 ∈ ℤ |
24 | dvdsnegb 15981 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 7 ∈ ℤ) → (2 ∥ 7 ↔ 2 ∥ -7)) | |
25 | 21, 23, 24 | mp2an 689 | . . . 4 ⊢ (2 ∥ 7 ↔ 2 ∥ -7) |
26 | 20, 25 | mtbi 322 | . . 3 ⊢ ¬ 2 ∥ -7 |
27 | znegcl 12355 | . . . 4 ⊢ (7 ∈ ℤ → -7 ∈ ℤ) | |
28 | mod2eq1n2dvds 16054 | . . . 4 ⊢ (-7 ∈ ℤ → ((-7 mod 2) = 1 ↔ ¬ 2 ∥ -7)) | |
29 | 23, 27, 28 | mp2b 10 | . . 3 ⊢ ((-7 mod 2) = 1 ↔ ¬ 2 ∥ -7) |
30 | 26, 29 | mpbir 230 | . 2 ⊢ (-7 mod 2) = 1 |
31 | 9, 30 | pm3.2i 471 | 1 ⊢ ((5 mod 3) = 2 ∧ (-7 mod 2) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 (class class class)co 7271 1c1 10873 + caddc 10875 < clt 11010 -cneg 11206 ℕcn 11973 2c2 12028 3c3 12029 5c5 12031 7c7 12033 ℕ0cn0 12233 ℤcz 12319 ℤ≥cuz 12581 mod cmo 13587 ∥ cdvds 15961 ℙcprime 16374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-ico 13084 df-fz 13239 df-fl 13510 df-mod 13588 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-dvds 15962 df-prm 16375 |
This theorem is referenced by: (None) |
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