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| Mirrors > Home > MPE Home > Th. List > phibndlem | Structured version Visualization version GIF version | ||
| Description: Lemma for phibnd 16808. (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 12924 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
| 2 | fzm1 13647 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑥 ∈ (1...𝑁) ↔ (𝑥 ∈ (1...(𝑁 − 1)) ∨ 𝑥 = 𝑁))) | |
| 3 | nnuz 12921 | . . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1) | |
| 4 | 2, 3 | eleq2s 2859 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (1...𝑁) ↔ (𝑥 ∈ (1...(𝑁 − 1)) ∨ 𝑥 = 𝑁))) |
| 5 | 4 | biimpa 476 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (1...𝑁)) → (𝑥 ∈ (1...(𝑁 − 1)) ∨ 𝑥 = 𝑁)) |
| 6 | 5 | ord 865 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (1...𝑁)) → (¬ 𝑥 ∈ (1...(𝑁 − 1)) → 𝑥 = 𝑁)) |
| 7 | 1, 6 | sylan 580 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑥 ∈ (1...𝑁)) → (¬ 𝑥 ∈ (1...(𝑁 − 1)) → 𝑥 = 𝑁)) |
| 8 | eluzelz 12888 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℤ) | |
| 9 | gcdid 16564 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 𝑁) = (abs‘𝑁)) | |
| 10 | 8, 9 | syl 17 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 gcd 𝑁) = (abs‘𝑁)) |
| 11 | nnre 12273 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
| 12 | nnnn0 12533 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
| 13 | 12 | nn0ge0d 12590 | . . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 0 ≤ 𝑁) |
| 14 | 11, 13 | absidd 15461 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (abs‘𝑁) = 𝑁) |
| 15 | 1, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → (abs‘𝑁) = 𝑁) |
| 16 | 10, 15 | eqtrd 2777 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 gcd 𝑁) = 𝑁) |
| 17 | 1re 11261 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
| 18 | eluz2gt1 12962 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘2) → 1 < 𝑁) | |
| 19 | ltne 11358 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 1 < 𝑁) → 𝑁 ≠ 1) | |
| 20 | 17, 18, 19 | sylancr 587 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ≠ 1) |
| 21 | 16, 20 | eqnetrd 3008 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 gcd 𝑁) ≠ 1) |
| 22 | oveq1 7438 | . . . . . . . 8 ⊢ (𝑥 = 𝑁 → (𝑥 gcd 𝑁) = (𝑁 gcd 𝑁)) | |
| 23 | 22 | neeq1d 3000 | . . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝑥 gcd 𝑁) ≠ 1 ↔ (𝑁 gcd 𝑁) ≠ 1)) |
| 24 | 21, 23 | syl5ibrcom 247 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑥 = 𝑁 → (𝑥 gcd 𝑁) ≠ 1)) |
| 25 | 24 | adantr 480 | . . . . 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 2960 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑥 ∈ (1...𝑁)) → ((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1...(𝑁 − 1)))) |
| 28 | 27 | ralrimiva 3146 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘2) → ∀𝑥 ∈ (1...𝑁)((𝑥 gcd 𝑁) = 1 → 𝑥 ∈ (1...(𝑁 − 1)))) |
| 29 | rabss 4072 | . 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 234 | 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 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 1c1 11156 < clt 11295 − cmin 11492 ℕcn 12266 2c2 12321 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 abscabs 15273 gcd cgcd 16531 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 |
| This theorem is referenced by: phibnd 16808 dfphi2 16811 |
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