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Mirrors > Home > MPE Home > Th. List > phibndlem | Structured version Visualization version GIF version |
Description: Lemma for phibnd 16739. (Contributed by Mario Carneiro, 23-Feb-2014.) |
Ref | Expression |
---|---|
phibndlem | ⊢ (𝑁 ∈ (ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2nn 12898 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
2 | fzm1 13613 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑥 ∈ (1...𝑁) ↔ (𝑥 ∈ (1...(𝑁 − 1)) ∨ 𝑥 = 𝑁))) | |
3 | nnuz 12895 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
4 | 2, 3 | eleq2s 2843 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (1...𝑁) ↔ (𝑥 ∈ (1...(𝑁 − 1)) ∨ 𝑥 = 𝑁))) |
5 | 4 | biimpa 475 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (1...𝑁)) → (𝑥 ∈ (1...(𝑁 − 1)) ∨ 𝑥 = 𝑁)) |
6 | 5 | ord 862 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (1...𝑁)) → (¬ 𝑥 ∈ (1...(𝑁 − 1)) → 𝑥 = 𝑁)) |
7 | 1, 6 | sylan 578 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑥 ∈ (1...𝑁)) → (¬ 𝑥 ∈ (1...(𝑁 − 1)) → 𝑥 = 𝑁)) |
8 | eluzelz 12862 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℤ) | |
9 | gcdid 16501 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 𝑁) = (abs‘𝑁)) | |
10 | 8, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 gcd 𝑁) = (abs‘𝑁)) |
11 | nnre 12249 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
12 | nnnn0 12509 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
13 | 12 | nn0ge0d 12565 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
14 | 11, 13 | absidd 15401 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (abs‘𝑁) = 𝑁) |
15 | 1, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (abs‘𝑁) = 𝑁) |
16 | 10, 15 | eqtrd 2765 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 gcd 𝑁) = 𝑁) |
17 | 1re 11244 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
18 | eluz2gt1 12934 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁) | |
19 | ltne 11341 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 1 < 𝑁) → 𝑁 ≠ 1) | |
20 | 17, 18, 19 | sylancr 585 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 1) |
21 | 16, 20 | eqnetrd 2998 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 gcd 𝑁) ≠ 1) |
22 | oveq1 7423 | . . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 gcd 𝑁) = (𝑁 gcd 𝑁)) | |
23 | 22 | neeq1d 2990 | . . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑥 gcd 𝑁) ≠ 1 ↔ (𝑁 gcd 𝑁) ≠ 1)) |
24 | 21, 23 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑥 = 𝑁 → (𝑥 gcd 𝑁) ≠ 1)) |
25 | 24 | adantr 479 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑥 ∈ (1...𝑁)) → (𝑥 = 𝑁 → (𝑥 gcd 𝑁) ≠ 1)) |
26 | 7, 25 | syld 47 | . . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑥 ∈ (1...𝑁)) → (¬ 𝑥 ∈ (1...(𝑁 − 1)) → (𝑥 gcd 𝑁) ≠ 1)) |
27 | 26 | necon4bd 2950 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑥 ∈ (1...𝑁)) → ((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1...(𝑁 − 1)))) |
28 | 27 | ralrimiva 3136 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ∀𝑥 ∈ (1...𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1...(𝑁 − 1)))) |
29 | rabss 4061 | . 2 ⊢ ({𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1)) ↔ ∀𝑥 ∈ (1...𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1...(𝑁 − 1)))) | |
30 | 28, 29 | sylibr 233 | 1 ⊢ (𝑁 ∈ (ℤ≥‘2) → {𝑥 ∈ (1...𝑁) ∣ (𝑥 gcd 𝑁) = 1} ⊆ (1...(𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 {crab 3419 ⊆ wss 3939 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 ℝcr 11137 1c1 11139 < clt 11278 − cmin 11474 ℕcn 12242 2c2 12297 ℤcz 12588 ℤ≥cuz 12852 ...cfz 13516 abscabs 15213 gcd cgcd 16468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-gcd 16469 |
This theorem is referenced by: phibnd 16739 dfphi2 16742 |
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