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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 9325 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | |
2 | zre 9286 | . . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
3 | ltne 8071 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
4 | 3 | necomd 2446 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵) |
5 | olc 712 | . . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
6 | dcne 2371 | . . . . . . . 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 713 | . . . . 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 9286 | . . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
16 | ltne 8071 | . . . . . . 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 1315 | . 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 709 DECID wdc 835 ∨ w3o 979 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 class class class wbr 4018 ℝcr 7839 < clt 8021 ℤcz 9282 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 |
This theorem is referenced by: nn0n0n1ge2b 9361 nn0lt2 9363 prime 9381 elnn1uz2 9636 iseqf1olemqcl 10516 iseqf1olemnab 10518 iseqf1olemab 10519 seq3f1olemstep 10531 exp3val 10552 hashfzp1 10835 fprod1p 11638 dvdsdc 11836 zdvdsdc 11850 dvdsabseq 11884 alzdvds 11891 fzo0dvdseq 11894 gcdmndc 11976 gcdsupex 11989 gcdsupcl 11990 gcd0id 12011 gcdaddm 12016 dfgcd2 12046 gcdmultiplez 12053 dvdssq 12063 nn0seqcvgd 12072 algcvgblem 12080 eucalgval2 12084 lcmmndc 12093 lcmdvds 12110 lcmid 12111 mulgcddvds 12125 cncongr2 12135 isprm3 12149 isprm4 12150 prm2orodd 12157 rpexp 12184 phivalfi 12243 phiprmpw 12253 phimullem 12256 eulerthlemfi 12259 hashgcdeq 12270 phisum 12271 pcxnn0cl 12341 pcge0 12344 pcdvdsb 12351 pcneg 12356 pcdvdstr 12358 pcgcd1 12359 pc2dvds 12361 pcz 12363 pcprmpw2 12364 pcmpt 12374 4sqlemafi 12426 4sqleminfi 12428 4sqexercise1 12429 4sqexercise2 12430 4sqlemsdc 12431 4sqlem11 12432 4sqlem19 12440 ennnfonelemim 12474 unbendc 12504 strsetsid 12544 mulgval 13061 mulgfng 13063 subgmulg 13124 lgsval 14858 lgsfvalg 14859 lgsfcl2 14860 lgscllem 14861 lgsval2lem 14864 lgsneg1 14879 lgsdir2 14887 lgsdirprm 14888 lgsdir 14889 lgsne0 14892 lgsprme0 14896 lgsdirnn0 14901 lgsdinn0 14902 2sqlem9 14924 nninffeq 15223 nconstwlpolem 15267 |
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