Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eulerthlemfi | GIF version |
Description: Lemma for eulerth 12096. The set 𝑆 is finite. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
Ref | Expression |
---|---|
eulerth.1 | ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) |
eulerth.2 | ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} |
Ref | Expression |
---|---|
eulerthlemfi | ⊢ (𝜑 → 𝑆 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 9172 | . . 3 ⊢ 0 ∈ ℤ | |
2 | eulerth.1 | . . . . 5 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) | |
3 | 2 | simp1d 994 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
4 | 3 | nnzd 9279 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
5 | fzofig 10324 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin) | |
6 | 1, 4, 5 | sylancr 411 | . 2 ⊢ (𝜑 → (0..^𝑁) ∈ Fin) |
7 | eulerth.2 | . . . 4 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} | |
8 | ssrab2 3213 | . . . 4 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁) | |
9 | 7, 8 | eqsstri 3160 | . . 3 ⊢ 𝑆 ⊆ (0..^𝑁) |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → 𝑆 ⊆ (0..^𝑁)) |
11 | elfzoelz 10039 | . . . . . . . 8 ⊢ (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ) | |
12 | 11 | adantl 275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℤ) |
13 | 4 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ) |
14 | 12, 13 | gcdcld 11843 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ0) |
15 | 14 | nn0zd 9278 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ) |
16 | 1zzd 9188 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → 1 ∈ ℤ) | |
17 | zdceq 9233 | . . . . 5 ⊢ (((𝑗 gcd 𝑁) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑗 gcd 𝑁) = 1) | |
18 | 15, 16, 17 | syl2anc 409 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → DECID (𝑗 gcd 𝑁) = 1) |
19 | oveq1 5828 | . . . . . . . . 9 ⊢ (𝑦 = 𝑗 → (𝑦 gcd 𝑁) = (𝑗 gcd 𝑁)) | |
20 | 19 | eqeq1d 2166 | . . . . . . . 8 ⊢ (𝑦 = 𝑗 → ((𝑦 gcd 𝑁) = 1 ↔ (𝑗 gcd 𝑁) = 1)) |
21 | 20, 7 | elrab2 2871 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑆 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 1)) |
22 | 21 | baibr 906 | . . . . . 6 ⊢ (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 1 ↔ 𝑗 ∈ 𝑆)) |
23 | 22 | dcbid 824 | . . . . 5 ⊢ (𝑗 ∈ (0..^𝑁) → (DECID (𝑗 gcd 𝑁) = 1 ↔ DECID 𝑗 ∈ 𝑆)) |
24 | 23 | adantl 275 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → (DECID (𝑗 gcd 𝑁) = 1 ↔ DECID 𝑗 ∈ 𝑆)) |
25 | 18, 24 | mpbid 146 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑁)) → DECID 𝑗 ∈ 𝑆) |
26 | 25 | ralrimiva 2530 | . 2 ⊢ (𝜑 → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆) |
27 | ssfidc 6876 | . 2 ⊢ (((0..^𝑁) ∈ Fin ∧ 𝑆 ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ 𝑆) → 𝑆 ∈ Fin) | |
28 | 6, 10, 26, 27 | syl3anc 1220 | 1 ⊢ (𝜑 → 𝑆 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 820 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 ∀wral 2435 {crab 2439 ⊆ wss 3102 (class class class)co 5821 Fincfn 6682 0cc0 7726 1c1 7727 ℕcn 8827 ℤcz 9161 ..^cfzo 10034 gcd cgcd 11821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-frec 6335 df-1o 6360 df-er 6477 df-en 6683 df-fin 6685 df-sup 6924 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-fz 9906 df-fzo 10035 df-fl 10162 df-mod 10215 df-seqfrec 10338 df-exp 10412 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-dvds 11677 df-gcd 11822 |
This theorem is referenced by: eulerthlemh 12094 eulerth 12096 |
Copyright terms: Public domain | W3C validator |