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Theorem phisum 12194
Description: The divisor sum identity of the totient function. Theorem 2.2 in [ApostolNT] p. 26. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Assertion
Ref Expression
phisum (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘𝑑) = 𝑁)
Distinct variable group:   𝑥,𝑁,𝑑

Proof of Theorem phisum
Dummy variables 𝑧 𝑦 𝑤 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3992 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑁𝑦𝑁))
21elrab 2886 . . . . 5 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↔ (𝑦 ∈ ℕ ∧ 𝑦𝑁))
3 hashgcdeq 12193 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = if(𝑦𝑁, (ϕ‘(𝑁 / 𝑦)), 0))
43adantrr 476 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦𝑁)) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = if(𝑦𝑁, (ϕ‘(𝑁 / 𝑦)), 0))
5 iftrue 3531 . . . . . . 7 (𝑦𝑁 → if(𝑦𝑁, (ϕ‘(𝑁 / 𝑦)), 0) = (ϕ‘(𝑁 / 𝑦)))
65ad2antll 488 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦𝑁)) → if(𝑦𝑁, (ϕ‘(𝑁 / 𝑦)), 0) = (ϕ‘(𝑁 / 𝑦)))
74, 6eqtrd 2203 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑦 ∈ ℕ ∧ 𝑦𝑁)) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (ϕ‘(𝑁 / 𝑦)))
82, 7sylan2b 285 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (ϕ‘(𝑁 / 𝑦)))
98sumeq2dv 11331 . . 3 (𝑁 ∈ ℕ → Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘(𝑁 / 𝑦)))
10 1zzd 9239 . . . . . 6 (𝑁 ∈ ℕ → 1 ∈ ℤ)
11 nnz 9231 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
1210, 11fzfigd 10387 . . . . 5 (𝑁 ∈ ℕ → (1...𝑁) ∈ Fin)
13 dvdsssfz1 11812 . . . . 5 (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ⊆ (1...𝑁))
14 elfznn 10010 . . . . . . . 8 (𝑗 ∈ (1...𝑁) → 𝑗 ∈ ℕ)
15 dvdsdc 11760 . . . . . . . 8 ((𝑗 ∈ ℕ ∧ 𝑁 ∈ ℤ) → DECID 𝑗𝑁)
1614, 11, 15syl2anr 288 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...𝑁)) → DECID 𝑗𝑁)
17 ibar 299 . . . . . . . . . . 11 (𝑗 ∈ ℕ → (𝑗𝑁 ↔ (𝑗 ∈ ℕ ∧ 𝑗𝑁)))
1814, 17syl 14 . . . . . . . . . 10 (𝑗 ∈ (1...𝑁) → (𝑗𝑁 ↔ (𝑗 ∈ ℕ ∧ 𝑗𝑁)))
19 breq1 3992 . . . . . . . . . . 11 (𝑥 = 𝑗 → (𝑥𝑁𝑗𝑁))
2019elrab 2886 . . . . . . . . . 10 (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↔ (𝑗 ∈ ℕ ∧ 𝑗𝑁))
2118, 20bitr4di 197 . . . . . . . . 9 (𝑗 ∈ (1...𝑁) → (𝑗𝑁𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}))
2221dcbid 833 . . . . . . . 8 (𝑗 ∈ (1...𝑁) → (DECID 𝑗𝑁DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}))
2322adantl 275 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...𝑁)) → (DECID 𝑗𝑁DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}))
2416, 23mpbid 146 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...𝑁)) → DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁})
2524ralrimiva 2543 . . . . 5 (𝑁 ∈ ℕ → ∀𝑗 ∈ (1...𝑁)DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁})
26 ssfidc 6912 . . . . 5 (((1...𝑁) ∈ Fin ∧ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ⊆ (1...𝑁) ∧ ∀𝑗 ∈ (1...𝑁)DECID 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∈ Fin)
2712, 13, 25, 26syl3anc 1233 . . . 4 (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥𝑁} ∈ Fin)
28 0z 9223 . . . . . . 7 0 ∈ ℤ
29 fzofig 10388 . . . . . . 7 ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0..^𝑁) ∈ Fin)
3028, 11, 29sylancr 412 . . . . . 6 (𝑁 ∈ ℕ → (0..^𝑁) ∈ Fin)
3130adantr 274 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (0..^𝑁) ∈ Fin)
32 ssrab2 3232 . . . . . 6 {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁)
3332a1i 9 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁))
34 elfzoelz 10103 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑁) → 𝑗 ∈ ℤ)
3534adantl 275 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℤ)
3611ad2antrr 485 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
3735, 36gcdcld 11923 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℕ0)
3837nn0zd 9332 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗 gcd 𝑁) ∈ ℤ)
39 elrabi 2883 . . . . . . . . . 10 (𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} → 𝑦 ∈ ℕ)
4039ad2antlr 486 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑦 ∈ ℕ)
4140nnzd 9333 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑦 ∈ ℤ)
42 zdceq 9287 . . . . . . . 8 (((𝑗 gcd 𝑁) ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID (𝑗 gcd 𝑁) = 𝑦)
4338, 41, 42syl2anc 409 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → DECID (𝑗 gcd 𝑁) = 𝑦)
44 ibar 299 . . . . . . . . . 10 (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 𝑦 ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 𝑦)))
45 oveq1 5860 . . . . . . . . . . . 12 (𝑧 = 𝑗 → (𝑧 gcd 𝑁) = (𝑗 gcd 𝑁))
4645eqeq1d 2179 . . . . . . . . . . 11 (𝑧 = 𝑗 → ((𝑧 gcd 𝑁) = 𝑦 ↔ (𝑗 gcd 𝑁) = 𝑦))
4746elrab 2886 . . . . . . . . . 10 (𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ↔ (𝑗 ∈ (0..^𝑁) ∧ (𝑗 gcd 𝑁) = 𝑦))
4844, 47bitr4di 197 . . . . . . . . 9 (𝑗 ∈ (0..^𝑁) → ((𝑗 gcd 𝑁) = 𝑦𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
4948dcbid 833 . . . . . . . 8 (𝑗 ∈ (0..^𝑁) → (DECID (𝑗 gcd 𝑁) = 𝑦DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
5049adantl 275 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → (DECID (𝑗 gcd 𝑁) = 𝑦DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
5143, 50mpbid 146 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) ∧ 𝑗 ∈ (0..^𝑁)) → DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
5251ralrimiva 2543 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
53 ssfidc 6912 . . . . 5 (((0..^𝑁) ∈ Fin ∧ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ⊆ (0..^𝑁) ∧ ∀𝑗 ∈ (0..^𝑁)DECID 𝑗 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ∈ Fin)
5431, 33, 52, 53syl3anc 1233 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ∈ Fin)
55 oveq1 5860 . . . . . . . . . 10 (𝑧 = 𝑤 → (𝑧 gcd 𝑁) = (𝑤 gcd 𝑁))
5655eqeq1d 2179 . . . . . . . . 9 (𝑧 = 𝑤 → ((𝑧 gcd 𝑁) = 𝑦 ↔ (𝑤 gcd 𝑁) = 𝑦))
5756elrab 2886 . . . . . . . 8 (𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} ↔ (𝑤 ∈ (0..^𝑁) ∧ (𝑤 gcd 𝑁) = 𝑦))
5857simprbi 273 . . . . . . 7 (𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} → (𝑤 gcd 𝑁) = 𝑦)
5958rgen 2523 . . . . . 6 𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦
6059rgenw 2525 . . . . 5 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦
61 invdisj 3983 . . . . 5 (∀𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}∀𝑤 ∈ {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} (𝑤 gcd 𝑁) = 𝑦Disj 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
6260, 61mp1i 10 . . . 4 (𝑁 ∈ ℕ → Disj 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦})
6327, 54, 62hashiun 11441 . . 3 (𝑁 ∈ ℕ → (♯‘ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (♯‘{𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
64 fveq2 5496 . . . 4 (𝑑 = (𝑁 / 𝑦) → (ϕ‘𝑑) = (ϕ‘(𝑁 / 𝑦)))
65 eqid 2170 . . . . 5 {𝑥 ∈ ℕ ∣ 𝑥𝑁} = {𝑥 ∈ ℕ ∣ 𝑥𝑁}
66 eqid 2170 . . . . 5 (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↦ (𝑁 / 𝑧)) = (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↦ (𝑁 / 𝑧))
6765, 66dvdsflip 11811 . . . 4 (𝑁 ∈ ℕ → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↦ (𝑁 / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥𝑁}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥𝑁})
68 oveq2 5861 . . . . 5 (𝑧 = 𝑦 → (𝑁 / 𝑧) = (𝑁 / 𝑦))
69 simpr 109 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁})
7011adantr 274 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → 𝑁 ∈ ℤ)
7139adantl 275 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → 𝑦 ∈ ℕ)
72 znq 9583 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑁 / 𝑦) ∈ ℚ)
7370, 71, 72syl2anc 409 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (𝑁 / 𝑦) ∈ ℚ)
7466, 68, 69, 73fvmptd3 5589 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↦ (𝑁 / 𝑧))‘𝑦) = (𝑁 / 𝑦))
75 elrabi 2883 . . . . . . 7 (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} → 𝑑 ∈ ℕ)
7675adantl 275 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → 𝑑 ∈ ℕ)
7776phicld 12172 . . . . 5 ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (ϕ‘𝑑) ∈ ℕ)
7877nncnd 8892 . . . 4 ((𝑁 ∈ ℕ ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁}) → (ϕ‘𝑑) ∈ ℂ)
7964, 27, 67, 74, 78fsumf1o 11353 . . 3 (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘𝑑) = Σ𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘(𝑁 / 𝑦)))
809, 63, 793eqtr4rd 2214 . 2 (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘𝑑) = (♯‘ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}))
81 iunrab 3920 . . . . 5 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦}
82 breq1 3992 . . . . . . . . 9 (𝑥 = (𝑧 gcd 𝑁) → (𝑥𝑁 ↔ (𝑧 gcd 𝑁) ∥ 𝑁))
83 elfzoelz 10103 . . . . . . . . . . 11 (𝑧 ∈ (0..^𝑁) → 𝑧 ∈ ℤ)
8483adantl 275 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → 𝑧 ∈ ℤ)
8511adantr 274 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
86 nnne0 8906 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → 𝑁 ≠ 0)
8786neneqd 2361 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → ¬ 𝑁 = 0)
8887intnand 926 . . . . . . . . . . 11 (𝑁 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑁 = 0))
8988adantr 274 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ¬ (𝑧 = 0 ∧ 𝑁 = 0))
90 gcdn0cl 11917 . . . . . . . . . 10 (((𝑧 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑁 = 0)) → (𝑧 gcd 𝑁) ∈ ℕ)
9184, 85, 89, 90syl21anc 1232 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∈ ℕ)
92 gcddvds 11918 . . . . . . . . . . 11 ((𝑧 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑧 gcd 𝑁) ∥ 𝑧 ∧ (𝑧 gcd 𝑁) ∥ 𝑁))
9384, 85, 92syl2anc 409 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ((𝑧 gcd 𝑁) ∥ 𝑧 ∧ (𝑧 gcd 𝑁) ∥ 𝑁))
9493simprd 113 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∥ 𝑁)
9582, 91, 94elrabd 2888 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → (𝑧 gcd 𝑁) ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁})
96 clel5 2867 . . . . . . . 8 ((𝑧 gcd 𝑁) ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} ↔ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦)
9795, 96sylib 121 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑧 ∈ (0..^𝑁)) → ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦)
9897ralrimiva 2543 . . . . . 6 (𝑁 ∈ ℕ → ∀𝑧 ∈ (0..^𝑁)∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦)
99 rabid2 2646 . . . . . 6 ((0..^𝑁) = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦} ↔ ∀𝑧 ∈ (0..^𝑁)∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦)
10098, 99sylibr 133 . . . . 5 (𝑁 ∈ ℕ → (0..^𝑁) = {𝑧 ∈ (0..^𝑁) ∣ ∃𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (𝑧 gcd 𝑁) = 𝑦})
10181, 100eqtr4id 2222 . . . 4 (𝑁 ∈ ℕ → 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦} = (0..^𝑁))
102101fveq2d 5500 . . 3 (𝑁 ∈ ℕ → (♯‘ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = (♯‘(0..^𝑁)))
103 nnnn0 9142 . . . 4 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
104 hashfzo0 10758 . . . 4 (𝑁 ∈ ℕ0 → (♯‘(0..^𝑁)) = 𝑁)
105103, 104syl 14 . . 3 (𝑁 ∈ ℕ → (♯‘(0..^𝑁)) = 𝑁)
106102, 105eqtrd 2203 . 2 (𝑁 ∈ ℕ → (♯‘ 𝑦 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} {𝑧 ∈ (0..^𝑁) ∣ (𝑧 gcd 𝑁) = 𝑦}) = 𝑁)
10780, 106eqtrd 2203 1 (𝑁 ∈ ℕ → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑁} (ϕ‘𝑑) = 𝑁)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  DECID wdc 829   = wceq 1348  wcel 2141  wral 2448  wrex 2449  {crab 2452  wss 3121  ifcif 3526   ciun 3873  Disj wdisj 3966   class class class wbr 3989  cmpt 4050  cfv 5198  (class class class)co 5853  Fincfn 6718  0cc0 7774  1c1 7775   / cdiv 8589  cn 8878  0cn0 9135  cz 9212  cq 9578  ...cfz 9965  ..^cfzo 10098  chash 10709  Σcsu 11316  cdvds 11749   gcd cgcd 11897  ϕcphi 12163
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-dvds 11750  df-gcd 11898  df-phi 12165
This theorem is referenced by: (None)
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