Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > odzcllem | Structured version Visualization version GIF version |
Description: - Lemma for odzcl 16130, showing existence of a recurrent point for the exponential. (Contributed by Mario Carneiro, 28-Feb-2014.) (Proof shortened by AV, 26-Sep-2020.) |
Ref | Expression |
---|---|
odzcllem | ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((odℤ‘𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((odℤ‘𝑁)‘𝐴)) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odzval 16128 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((odℤ‘𝑁)‘𝐴) = inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < )) | |
2 | ssrab2 4056 | . . . . 5 ⊢ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ⊆ ℕ | |
3 | nnuz 12282 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | sseqtri 4003 | . . . 4 ⊢ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ⊆ (ℤ≥‘1) |
5 | phicl 16106 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (ϕ‘𝑁) ∈ ℕ) | |
6 | 5 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ ℕ) |
7 | eulerth 16120 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁)) | |
8 | simp1 1132 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∈ ℕ) | |
9 | simp2 1133 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝐴 ∈ ℤ) | |
10 | 6 | nnnn0d 11956 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (ϕ‘𝑁) ∈ ℕ0) |
11 | zexpcl 13445 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑁) ∈ ℕ0) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ) | |
12 | 9, 10, 11 | syl2anc 586 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ) |
13 | 1z 12013 | . . . . . . . . 9 ⊢ 1 ∈ ℤ | |
14 | moddvds 15618 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑁)) ∈ ℤ ∧ 1 ∈ ℤ) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))) | |
15 | 13, 14 | mp3an3 1446 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑁)) ∈ ℤ) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))) |
16 | 8, 12, 15 | syl2anc 586 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))) |
17 | 7, 16 | mpbid 234 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)) |
18 | oveq2 7164 | . . . . . . . . 9 ⊢ (𝑛 = (ϕ‘𝑁) → (𝐴↑𝑛) = (𝐴↑(ϕ‘𝑁))) | |
19 | 18 | oveq1d 7171 | . . . . . . . 8 ⊢ (𝑛 = (ϕ‘𝑁) → ((𝐴↑𝑛) − 1) = ((𝐴↑(ϕ‘𝑁)) − 1)) |
20 | 19 | breq2d 5078 | . . . . . . 7 ⊢ (𝑛 = (ϕ‘𝑁) → (𝑁 ∥ ((𝐴↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))) |
21 | 20 | rspcev 3623 | . . . . . 6 ⊢ (((ϕ‘𝑁) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)) → ∃𝑛 ∈ ℕ 𝑁 ∥ ((𝐴↑𝑛) − 1)) |
22 | 6, 17, 21 | syl2anc 586 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ∃𝑛 ∈ ℕ 𝑁 ∥ ((𝐴↑𝑛) − 1)) |
23 | rabn0 4339 | . . . . 5 ⊢ ({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ≠ ∅ ↔ ∃𝑛 ∈ ℕ 𝑁 ∥ ((𝐴↑𝑛) − 1)) | |
24 | 22, 23 | sylibr 236 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ≠ ∅) |
25 | infssuzcl 12333 | . . . 4 ⊢ (({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ⊆ (ℤ≥‘1) ∧ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ≠ ∅) → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}) | |
26 | 4, 24, 25 | sylancr 589 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → inf({𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}, ℝ, < ) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}) |
27 | 1, 26 | eqeltrd 2913 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((odℤ‘𝑁)‘𝐴) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)}) |
28 | oveq2 7164 | . . . . 5 ⊢ (𝑛 = ((odℤ‘𝑁)‘𝐴) → (𝐴↑𝑛) = (𝐴↑((odℤ‘𝑁)‘𝐴))) | |
29 | 28 | oveq1d 7171 | . . . 4 ⊢ (𝑛 = ((odℤ‘𝑁)‘𝐴) → ((𝐴↑𝑛) − 1) = ((𝐴↑((odℤ‘𝑁)‘𝐴)) − 1)) |
30 | 29 | breq2d 5078 | . . 3 ⊢ (𝑛 = ((odℤ‘𝑁)‘𝐴) → (𝑁 ∥ ((𝐴↑𝑛) − 1) ↔ 𝑁 ∥ ((𝐴↑((odℤ‘𝑁)‘𝐴)) − 1))) |
31 | 30 | elrab 3680 | . 2 ⊢ (((odℤ‘𝑁)‘𝐴) ∈ {𝑛 ∈ ℕ ∣ 𝑁 ∥ ((𝐴↑𝑛) − 1)} ↔ (((odℤ‘𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((odℤ‘𝑁)‘𝐴)) − 1))) |
32 | 27, 31 | sylib 220 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → (((odℤ‘𝑁)‘𝐴) ∈ ℕ ∧ 𝑁 ∥ ((𝐴↑((odℤ‘𝑁)‘𝐴)) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 ⊆ wss 3936 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 infcinf 8905 ℝcr 10536 1c1 10538 < clt 10675 − cmin 10870 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 ℤ≥cuz 12244 mod cmo 13238 ↑cexp 13430 ∥ cdvds 15607 gcd cgcd 15843 odℤcodz 16100 ϕcphi 16101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-odz 16102 df-phi 16103 |
This theorem is referenced by: odzcl 16130 odzid 16131 |
Copyright terms: Public domain | W3C validator |