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| Mirrors > Home > MPE Home > Th. List > ex-mod | Structured version Visualization version GIF version | ||
| Description: Example for df-mod 13902. (Contributed by AV, 3-Sep-2021.) |
| Ref | Expression |
|---|---|
| ex-mod | ⊢ ((5 mod 3) = 2 ∧ (-7 mod 2) = 1) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3p2e5 12390 | . . . . 5 ⊢ (3 + 2) = 5 | |
| 2 | 1 | eqcomi 2770 | . . . 4 ⊢ 5 = (3 + 2) |
| 3 | 2 | oveq1i 7420 | . . 3 ⊢ (5 mod 3) = ((3 + 2) mod 3) |
| 4 | 2nn0 12520 | . . . 4 ⊢ 2 ∈ ℕ0 | |
| 5 | 3nn 12319 | . . . 4 ⊢ 3 ∈ ℕ | |
| 6 | 2lt3 12413 | . . . 4 ⊢ 2 < 3 | |
| 7 | addmodid 13954 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 3 ∈ ℕ ∧ 2 < 3) → ((3 + 2) mod 3) = 2) | |
| 8 | 4, 5, 6, 7 | mp3an 1488 | . . 3 ⊢ ((3 + 2) mod 3) = 2 |
| 9 | 3, 8 | eqtri 2784 | . 2 ⊢ (5 mod 3) = 2 |
| 10 | 2re 12314 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 11 | 2lt7 12432 | . . . . . 6 ⊢ 2 < 7 | |
| 12 | 10, 11 | ltneii 11322 | . . . . 5 ⊢ 2 ≠ 7 |
| 13 | 2nn 12313 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
| 14 | 1lt2 12412 | . . . . . . 7 ⊢ 1 < 2 | |
| 15 | eluz2b2 12944 | . . . . . . 7 ⊢ (2 ∈ (ℤ≥‘2) ↔ (2 ∈ ℕ ∧ 1 < 2)) | |
| 16 | 13, 14, 15 | mpbir2an 723 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
| 17 | 7prm 17169 | . . . . . 6 ⊢ 7 ∈ ℙ | |
| 18 | dvdsprm 16761 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 7 ∈ ℙ) → (2 ∥ 7 ↔ 2 = 7)) | |
| 19 | 16, 17, 18 | mp2an 704 | . . . . 5 ⊢ (2 ∥ 7 ↔ 2 = 7) |
| 20 | 12, 19 | nemtbir 3052 | . . . 4 ⊢ ¬ 2 ∥ 7 |
| 21 | 2z 12625 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 22 | 7nn 12332 | . . . . . 6 ⊢ 7 ∈ ℕ | |
| 23 | 22 | nnzi 12617 | . . . . 5 ⊢ 7 ∈ ℤ |
| 24 | dvdsnegb 16330 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 7 ∈ ℤ) → (2 ∥ 7 ↔ 2 ∥ -7)) | |
| 25 | 21, 23, 24 | mp2an 704 | . . . 4 ⊢ (2 ∥ 7 ↔ 2 ∥ -7) |
| 26 | 20, 25 | mtbi 325 | . . 3 ⊢ ¬ 2 ∥ -7 |
| 27 | znegcl 12628 | . . . 4 ⊢ (7 ∈ ℤ → -7 ∈ ℤ) | |
| 28 | mod2eq1n2dvds 16404 | . . . 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 234 | . 2 ⊢ (-7 mod 2) = 1 |
| 31 | 9, 30 | pm3.2i 475 | 1 ⊢ ((5 mod 3) = 2 ∧ (-7 mod 2) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 400 = wceq 1568 ∈ wcel 2141 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 1c1 11100 + caddc 11102 < clt 11242 -cneg 11441 ℕcn 12232 2c2 12294 3c3 12295 5c5 12297 7c7 12299 ℕ0cn0 12503 ℤcz 12590 ℤ≥cuz 12861 mod cmo 13901 ∥ cdvds 16309 ℙcprime 16728 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 |
| This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-rp 13016 df-ico 13377 df-fz 13535 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-dvds 16310 df-prm 16729 |
| This theorem is referenced by: (None) |
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