zdceq - Intuitionistic Logic Explorer

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Theorem zdceq 9447

Description: Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.)

**Assertion**
Ref	Expression
zdceq	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID 𝐴 = 𝐵)

**Proof of Theorem zdceq**
Step	Hyp	Ref	Expression
1		ztri3or 9414	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
2		zre 9375	. . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
3		ltne 8156	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴)
4	3	necomd 2461	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵)
5		olc 712	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵))
6		dcne 2386	. . . . . . . 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 9375	. . . . 5 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
16		ltne 8156	. . . . . . 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 1316	. 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 1372 ∈ wcel 2175 ≠ wne 2375 class class class wbr 4043 ℝcr 7923 < clt 8106 ℤcz 9371

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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-addcom 8024 ax-addass 8026 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-0id 8032 ax-rnegex 8033 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-ltadd 8040

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-inn 9036 df-n0 9295 df-z 9372

This theorem is referenced by: nn0n0n1ge2b 9451 nn0lt2 9453 prime 9471 elnn1uz2 9727 iseqf1olemqcl 10642 iseqf1olemnab 10644 iseqf1olemab 10645 seq3f1olemstep 10657 exp3val 10684 hashfzp1 10967 ccat1st1st 11091 fprod1p 11881 dvdsdc 12080 zdvdsdc 12094 fsumdvds 12124 dvdsabseq 12129 alzdvds 12136 fzo0dvdseq 12139 gcdmndc 12247 gcdsupex 12249 gcdsupcl 12250 gcd0id 12271 gcdaddm 12276 dfgcd2 12306 gcdmultiplez 12313 dvdssq 12323 nn0seqcvgd 12334 algcvgblem 12342 eucalgval2 12346 lcmmndc 12355 lcmdvds 12372 lcmid 12373 mulgcddvds 12387 cncongr2 12397 isprm3 12411 isprm4 12412 prm2orodd 12419 rpexp 12446 phivalfi 12505 phiprmpw 12515 phimullem 12518 eulerthlemfi 12521 hashgcdeq 12533 phisum 12534 pcxnn0cl 12604 pcge0 12607 pcdvdsb 12614 pcneg 12619 pcdvdstr 12621 pcgcd1 12622 pc2dvds 12624 pcz 12626 pcprmpw2 12627 pcmpt 12637 4sqlemafi 12689 4sqleminfi 12691 4sqexercise1 12692 4sqexercise2 12693 4sqlemsdc 12694 4sqlem11 12695 4sqlem19 12703 ennnfonelemim 12766 unbendc 12796 strsetsid 12836 mulgval 13429 mulgfng 13431 subgmulg 13495 znf1o 14384 psr1clfi 14421 ply1term 15186 dvply1 15208 perfectlem2 15443 lgsval 15452 lgsfvalg 15453 lgsfcl2 15454 lgscllem 15455 lgsval2lem 15458 lgsneg1 15473 lgsdir2 15481 lgsdirprm 15482 lgsdir 15483 lgsne0 15486 lgsprme0 15490 lgsdirnn0 15495 lgsdinn0 15496 lgsquadlem1 15525 lgsquadlem2 15526 lgsquad3 15532 2lgs 15552 2lgsoddprm 15561 2sqlem9 15572 nninffeq 15919 nconstwlpolem 15966

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