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Mirrors > Home > ILE Home > Th. List > zdceq | GIF version |
Description: Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zdceq | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ztri3or 8854 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | |
2 | zre 8815 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | ltne 7631 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
4 | 3 | necomd 2342 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵) |
5 | olc 668 | . . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
6 | dcne 2267 | . . . . . . . 8 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
7 | 5, 6 | sylibr 133 | . . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → DECID 𝐴 = 𝐵) |
8 | 4, 7 | syl 14 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → DECID 𝐴 = 𝐵) |
9 | 8 | ex 114 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
10 | 9 | adantr 271 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
11 | 2, 10 | sylan 278 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
12 | orc 669 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
13 | 12, 6 | sylibr 133 | . . . 4 ⊢ (𝐴 = 𝐵 → DECID 𝐴 = 𝐵) |
14 | 13 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 = 𝐵 → DECID 𝐴 = 𝐵)) |
15 | zre 8815 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
16 | ltne 7631 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → 𝐴 ≠ 𝐵) | |
17 | 16, 7 | syl 14 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → DECID 𝐴 = 𝐵) |
18 | 17 | ex 114 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
19 | 15, 18 | syl 14 | . . . 4 ⊢ (𝐵 ∈ ℤ → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
20 | 19 | adantl 272 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
21 | 11, 14, 20 | 3jaod 1241 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 = 𝐵)) |
22 | 1, 21 | mpd 13 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 665 DECID wdc 781 ∨ w3o 924 = wceq 1290 ∈ wcel 1439 ≠ wne 2256 class class class wbr 3851 ℝcr 7410 < clt 7583 ℤcz 8811 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-inn 8484 df-n0 8735 df-z 8812 |
This theorem is referenced by: nn0n0n1ge2b 8887 nn0lt2 8889 prime 8906 elnn1uz2 9155 iseqf1olemqcl 9976 iseqf1olemnab 9978 iseqf1olemab 9979 seq3f1olemstep 9991 exp3val 10018 hashfzp1 10293 dvdsdc 11143 zdvdsdc 11156 dvdsabseq 11187 alzdvds 11194 fzo0dvdseq 11197 gcdmndc 11279 gcdsupex 11288 gcdsupcl 11289 gcd0id 11309 gcdaddm 11314 dfgcd2 11342 gcdmultiplez 11349 dvdssq 11359 nn0seqcvgd 11362 algcvgblem 11370 eucalgval2 11374 lcmmndc 11383 lcmdvds 11400 lcmid 11401 mulgcddvds 11415 cncongr2 11425 isprm3 11439 isprm4 11440 prm2orodd 11447 rpexp 11471 phivalfi 11527 phiprmpw 11537 phimullem 11540 hashgcdeq 11543 strsetsid 11588 |
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