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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 9285 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | |
2 | zre 9246 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | ltne 8032 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
4 | 3 | necomd 2433 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵) |
5 | olc 711 | . . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
6 | dcne 2358 | . . . . . . . 8 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
7 | 5, 6 | sylibr 134 | . . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → DECID 𝐴 = 𝐵) |
8 | 4, 7 | syl 14 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → DECID 𝐴 = 𝐵) |
9 | 8 | ex 115 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
10 | 9 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
11 | 2, 10 | sylan 283 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
12 | orc 712 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
13 | 12, 6 | sylibr 134 | . . . 4 ⊢ (𝐴 = 𝐵 → DECID 𝐴 = 𝐵) |
14 | 13 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 = 𝐵 → DECID 𝐴 = 𝐵)) |
15 | zre 9246 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
16 | ltne 8032 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → 𝐴 ≠ 𝐵) | |
17 | 16, 7 | syl 14 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → DECID 𝐴 = 𝐵) |
18 | 17 | ex 115 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
19 | 15, 18 | syl 14 | . . . 4 ⊢ (𝐵 ∈ ℤ → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
20 | 19 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
21 | 11, 14, 20 | 3jaod 1304 | . 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 104 ∨ wo 708 DECID wdc 834 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 ℝcr 7801 < clt 7982 ℤcz 9242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-n0 9166 df-z 9243 |
This theorem is referenced by: nn0n0n1ge2b 9321 nn0lt2 9323 prime 9341 elnn1uz2 9596 iseqf1olemqcl 10472 iseqf1olemnab 10474 iseqf1olemab 10475 seq3f1olemstep 10487 exp3val 10508 hashfzp1 10788 fprod1p 11591 dvdsdc 11789 zdvdsdc 11803 dvdsabseq 11836 alzdvds 11843 fzo0dvdseq 11846 gcdmndc 11928 gcdsupex 11941 gcdsupcl 11942 gcd0id 11963 gcdaddm 11968 dfgcd2 11998 gcdmultiplez 12005 dvdssq 12015 nn0seqcvgd 12024 algcvgblem 12032 eucalgval2 12036 lcmmndc 12045 lcmdvds 12062 lcmid 12063 mulgcddvds 12077 cncongr2 12087 isprm3 12101 isprm4 12102 prm2orodd 12109 rpexp 12136 phivalfi 12195 phiprmpw 12205 phimullem 12208 eulerthlemfi 12211 hashgcdeq 12222 phisum 12223 pcxnn0cl 12293 pcge0 12295 pcdvdsb 12302 pcneg 12307 pcdvdstr 12309 pcgcd1 12310 pc2dvds 12312 pcz 12314 pcprmpw2 12315 pcmpt 12324 ennnfonelemim 12408 unbendc 12438 strsetsid 12478 mulgval 12875 mulgfng 12876 lgsval 14072 lgsfvalg 14073 lgsfcl2 14074 lgscllem 14075 lgsval2lem 14078 lgsneg1 14093 lgsdir2 14101 lgsdirprm 14102 lgsdir 14103 lgsne0 14106 lgsprme0 14110 lgsdirnn0 14115 lgsdinn0 14116 2sqlem9 14127 nninffeq 14425 nconstwlpolem 14468 |
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