odzcllem - Intuitionistic Logic Explorer

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Theorem odzcllem 12411

Description: - Lemma for odzcl 12412, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.)

**Assertion**
Ref	Expression
odzcllem	⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((od_ℤ‘𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((od_ℤ‘𝑁)‘𝐴)) − 1)))

Proof of Theorem odzcllem Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		odzval 12410	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od_ℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ))
2		1zzd 9353	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 1 ∈ ℤ)
3		nnuz 9637	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
4	3	rabeqi 2756	. . . 4 ⊢ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} = {𝑛 ∈ (ℤ_≥‘1) ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}
5		oveq2 5930	. . . . . . 7 ⊢ (𝑛 = (ϕ‘𝑁) → (𝐴↑𝑛) = (𝐴↑(ϕ‘𝑁)))
6	5	oveq1d 5937	. . . . . 6 ⊢ (𝑛 = (ϕ‘𝑁) → ((𝐴↑𝑛) − 1) = ((𝐴↑(ϕ‘𝑁)) − 1))
7	6	breq2d 4045	. . . . 5 ⊢ (𝑛 = (ϕ‘𝑁) → (𝑁 ∥ ((𝐴↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
8		phicl 12383	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ)
9	8	3ad2ant1 1020	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ ℕ)
10		eulerth 12401	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))
11		simp1 999	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℕ)
12		simp2 1000	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ∈ ℤ)
13	9	nnnn0d 9302	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ ℕ₀)
14		zexpcl 10646	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
15	12, 13, 14	syl2anc 411	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
16		1z 9352	. . . . . . . 8 ⊢ 1 ∈ ℤ
17		moddvds 11964	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑁)) ∈ ℤ ∧ 1 ∈ ℤ) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
18	16, 17	mp3an3 1337	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑁)) ∈ ℤ) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
19	11, 15, 18	syl2anc 411	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
20	10, 19	mpbid 147	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))
21	7, 9, 20	elrabd 2922	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
22		elfznn 10129	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(ϕ‘𝑁)) → 𝑛 ∈ ℕ)
23	22	adantl 277	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑛 ∈ (1...(ϕ‘𝑁))) → 𝑛 ∈ ℕ)
24	23	nnnn0d 9302	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑛 ∈ (1...(ϕ‘𝑁))) → 𝑛 ∈ ℕ₀)
25		zexpcl 10646	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → (𝐴↑𝑛) ∈ ℤ)
26	12, 24, 25	syl2an2r 595	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑛 ∈ (1...(ϕ‘𝑁))) → (𝐴↑𝑛) ∈ ℤ)
27		peano2zm 9364	. . . . . 6 ⊢ ((𝐴↑𝑛) ∈ ℤ → ((𝐴↑𝑛) − 1) ∈ ℤ)
28	26, 27	syl 14	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑛 ∈ (1...(ϕ‘𝑁))) → ((𝐴↑𝑛) − 1) ∈ ℤ)
29		dvdsdc 11963	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴↑𝑛) − 1) ∈ ℤ) → DECID 𝑁 ∥ ((𝐴↑𝑛) − 1))
30	11, 28, 29	syl2an2r 595	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) ∧ 𝑛 ∈ (1...(ϕ‘𝑁))) → DECID 𝑁 ∥ ((𝐴↑𝑛) − 1))
31	2, 4, 21, 30	infssuzcldc 10325	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
32	1, 31	eqeltrd 2273	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((od_ℤ‘𝑁)‘𝐴) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)})
33		oveq2 5930	. . . . 5 ⊢ (𝑛 = ((od_ℤ‘𝑁)‘𝐴) → (𝐴↑𝑛) = (𝐴↑((od_ℤ‘𝑁)‘𝐴)))
34	33	oveq1d 5937	. . . 4 ⊢ (𝑛 = ((od_ℤ‘𝑁)‘𝐴) → ((𝐴↑𝑛) − 1) = ((𝐴↑((od_ℤ‘𝑁)‘𝐴)) − 1))
35	34	breq2d 4045	. . 3 ⊢ (𝑛 = ((od_ℤ‘𝑁)‘𝐴) → (𝑁 ∥ ((𝐴↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑((od_ℤ‘𝑁)‘𝐴)) − 1)))
36	35	elrab 2920	. 2 ⊢ (((od_ℤ‘𝑁)‘𝐴) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ↔ (((od_ℤ‘𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((od_ℤ‘𝑁)‘𝐴)) − 1)))
37	32, 36	sylib 122	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((od_ℤ‘𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((od_ℤ‘𝑁)‘𝐴)) − 1)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 {crab 2479 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 infcinf 7049 ℝcr 7878 1c1 7880 < clt 8061 − cmin 8197 ℕcn 8990 ℕ₀cn0 9249 ℤcz 9326 ℤ_≥cuz 9601 ...cfz 10083 mod cmo 10414 ↑cexp 10630 ∥ cdvds 11952 gcd cgcd 12120 od_ℤcodz 12376 ϕcphi 12377

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-proddc 11716 df-dvds 11953 df-gcd 12121 df-odz 12378 df-phi 12379

This theorem is referenced by: odzcl 12412 odzid 12413

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