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 fprod1p 11852 dvdsdc 12051 zdvdsdc 12065 fsumdvds 12095 dvdsabseq 12100 alzdvds 12107 fzo0dvdseq 12110 gcdmndc 12218 gcdsupex 12220 gcdsupcl 12221 gcd0id 12242 gcdaddm 12247 dfgcd2 12277 gcdmultiplez 12284 dvdssq 12294 nn0seqcvgd 12305 algcvgblem 12313 eucalgval2 12317 lcmmndc 12326 lcmdvds 12343 lcmid 12344 mulgcddvds 12358 cncongr2 12368 isprm3 12382 isprm4 12383 prm2orodd 12390 rpexp 12417 phivalfi 12476 phiprmpw 12486 phimullem 12489 eulerthlemfi 12492 hashgcdeq 12504 phisum 12505 pcxnn0cl 12575 pcge0 12578 pcdvdsb 12585 pcneg 12590 pcdvdstr 12592 pcgcd1 12593 pc2dvds 12595 pcz 12597 pcprmpw2 12598 pcmpt 12608 4sqlemafi 12660 4sqleminfi 12662 4sqexercise1 12663 4sqexercise2 12664 4sqlemsdc 12665 4sqlem11 12666 4sqlem19 12674 ennnfonelemim 12737 unbendc 12767 strsetsid 12807 mulgval 13400 mulgfng 13402 subgmulg 13466 znf1o 14355 psr1clfi 14392 ply1term 15157 dvply1 15179 perfectlem2 15414 lgsval 15423 lgsfvalg 15424 lgsfcl2 15425 lgscllem 15426 lgsval2lem 15429 lgsneg1 15444 lgsdir2 15452 lgsdirprm 15453 lgsdir 15454 lgsne0 15457 lgsprme0 15461 lgsdirnn0 15466 lgsdinn0 15467 lgsquadlem1 15496 lgsquadlem2 15497 lgsquad3 15503 2lgs 15523 2lgsoddprm 15532 2sqlem9 15543 nninffeq 15890 nconstwlpolem 15937

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