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Theorem eulerthlemrprm 12183
Description: Lemma for eulerth 12187. 𝑁 and 𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥) are relatively prime. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.)
Hypotheses
Ref Expression
eulerth.1 (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
eulerth.2 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
eulerth.4 (𝜑𝐹:(1...(ϕ‘𝑁))–1-1-onto𝑆)
Assertion
Ref Expression
eulerthlemrprm (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)) = 1)
Distinct variable groups:   𝑥,𝐹   𝑦,𝐹   𝑥,𝑁   𝑦,𝑁   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥,𝑦)   𝑆(𝑥,𝑦)

Proof of Theorem eulerthlemrprm
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eulerth.1 . . . . . 6 (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
21simp1d 1004 . . . . 5 (𝜑𝑁 ∈ ℕ)
32phicld 12172 . . . 4 (𝜑 → (ϕ‘𝑁) ∈ ℕ)
4 elnnuz 9523 . . . 4 ((ϕ‘𝑁) ∈ ℕ ↔ (ϕ‘𝑁) ∈ (ℤ‘1))
53, 4sylib 121 . . 3 (𝜑 → (ϕ‘𝑁) ∈ (ℤ‘1))
6 eluzfz2 9988 . . 3 ((ϕ‘𝑁) ∈ (ℤ‘1) → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
75, 6syl 14 . 2 (𝜑 → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
8 oveq2 5861 . . . . . . 7 (𝑤 = 1 → (1...𝑤) = (1...1))
98prodeq1d 11527 . . . . . 6 (𝑤 = 1 → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...1)(𝐹𝑥))
109oveq2d 5869 . . . . 5 (𝑤 = 1 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)))
1110eqeq1d 2179 . . . 4 (𝑤 = 1 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)) = 1))
1211imbi2d 229 . . 3 (𝑤 = 1 → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)) = 1)))
13 oveq2 5861 . . . . . . 7 (𝑤 = 𝑘 → (1...𝑤) = (1...𝑘))
1413prodeq1d 11527 . . . . . 6 (𝑤 = 𝑘 → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...𝑘)(𝐹𝑥))
1514oveq2d 5869 . . . . 5 (𝑤 = 𝑘 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)))
1615eqeq1d 2179 . . . 4 (𝑤 = 𝑘 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1))
1716imbi2d 229 . . 3 (𝑤 = 𝑘 → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1)))
18 oveq2 5861 . . . . . . 7 (𝑤 = (𝑘 + 1) → (1...𝑤) = (1...(𝑘 + 1)))
1918prodeq1d 11527 . . . . . 6 (𝑤 = (𝑘 + 1) → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥))
2019oveq2d 5869 . . . . 5 (𝑤 = (𝑘 + 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)))
2120eqeq1d 2179 . . . 4 (𝑤 = (𝑘 + 1) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1))
2221imbi2d 229 . . 3 (𝑤 = (𝑘 + 1) → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1)))
23 oveq2 5861 . . . . . . 7 (𝑤 = (ϕ‘𝑁) → (1...𝑤) = (1...(ϕ‘𝑁)))
2423prodeq1d 11527 . . . . . 6 (𝑤 = (ϕ‘𝑁) → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥))
2524oveq2d 5869 . . . . 5 (𝑤 = (ϕ‘𝑁) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)))
2625eqeq1d 2179 . . . 4 (𝑤 = (ϕ‘𝑁) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)) = 1))
2726imbi2d 229 . . 3 (𝑤 = (ϕ‘𝑁) → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)) = 1)))
28 1z 9238 . . . . . . 7 1 ∈ ℤ
29 eulerth.2 . . . . . . . . . . 11 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
30 ssrab2 3232 . . . . . . . . . . 11 {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
3129, 30eqsstri 3179 . . . . . . . . . 10 𝑆 ⊆ (0..^𝑁)
32 fzo0ssnn0 10171 . . . . . . . . . 10 (0..^𝑁) ⊆ ℕ0
3331, 32sstri 3156 . . . . . . . . 9 𝑆 ⊆ ℕ0
34 nn0sscn 9140 . . . . . . . . 9 0 ⊆ ℂ
3533, 34sstri 3156 . . . . . . . 8 𝑆 ⊆ ℂ
36 eulerth.4 . . . . . . . . . 10 (𝜑𝐹:(1...(ϕ‘𝑁))–1-1-onto𝑆)
37 f1of 5442 . . . . . . . . . 10 (𝐹:(1...(ϕ‘𝑁))–1-1-onto𝑆𝐹:(1...(ϕ‘𝑁))⟶𝑆)
3836, 37syl 14 . . . . . . . . 9 (𝜑𝐹:(1...(ϕ‘𝑁))⟶𝑆)
393nnge1d 8921 . . . . . . . . . 10 (𝜑 → 1 ≤ (ϕ‘𝑁))
40 uzid 9501 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
4128, 40ax-mp 5 . . . . . . . . . . 11 1 ∈ (ℤ‘1)
423nnzd 9333 . . . . . . . . . . 11 (𝜑 → (ϕ‘𝑁) ∈ ℤ)
43 elfz5 9973 . . . . . . . . . . 11 ((1 ∈ (ℤ‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
4441, 42, 43sylancr 412 . . . . . . . . . 10 (𝜑 → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
4539, 44mpbird 166 . . . . . . . . 9 (𝜑 → 1 ∈ (1...(ϕ‘𝑁)))
4638, 45ffvelrnd 5632 . . . . . . . 8 (𝜑 → (𝐹‘1) ∈ 𝑆)
4735, 46sselid 3145 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℂ)
48 fveq2 5496 . . . . . . . 8 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
4948fprod1 11557 . . . . . . 7 ((1 ∈ ℤ ∧ (𝐹‘1) ∈ ℂ) → ∏𝑥 ∈ (1...1)(𝐹𝑥) = (𝐹‘1))
5028, 47, 49sylancr 412 . . . . . 6 (𝜑 → ∏𝑥 ∈ (1...1)(𝐹𝑥) = (𝐹‘1))
5150oveq2d 5869 . . . . 5 (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)) = (𝑁 gcd (𝐹‘1)))
522nnzd 9333 . . . . . 6 (𝜑𝑁 ∈ ℤ)
53 nn0ssz 9230 . . . . . . . 8 0 ⊆ ℤ
5433, 53sstri 3156 . . . . . . 7 𝑆 ⊆ ℤ
5554, 46sselid 3145 . . . . . 6 (𝜑 → (𝐹‘1) ∈ ℤ)
56 gcdcom 11928 . . . . . 6 ((𝑁 ∈ ℤ ∧ (𝐹‘1) ∈ ℤ) → (𝑁 gcd (𝐹‘1)) = ((𝐹‘1) gcd 𝑁))
5752, 55, 56syl2anc 409 . . . . 5 (𝜑 → (𝑁 gcd (𝐹‘1)) = ((𝐹‘1) gcd 𝑁))
58 oveq1 5860 . . . . . . . . 9 (𝑦 = (𝐹‘1) → (𝑦 gcd 𝑁) = ((𝐹‘1) gcd 𝑁))
5958eqeq1d 2179 . . . . . . . 8 (𝑦 = (𝐹‘1) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘1) gcd 𝑁) = 1))
6059, 29elrab2 2889 . . . . . . 7 ((𝐹‘1) ∈ 𝑆 ↔ ((𝐹‘1) ∈ (0..^𝑁) ∧ ((𝐹‘1) gcd 𝑁) = 1))
6146, 60sylib 121 . . . . . 6 (𝜑 → ((𝐹‘1) ∈ (0..^𝑁) ∧ ((𝐹‘1) gcd 𝑁) = 1))
6261simprd 113 . . . . 5 (𝜑 → ((𝐹‘1) gcd 𝑁) = 1)
6351, 57, 623eqtrd 2207 . . . 4 (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)) = 1)
6463a1i 9 . . 3 ((ϕ‘𝑁) ∈ (ℤ‘1) → (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)) = 1))
65 simpr 109 . . . . . . . 8 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1)
6638adantr 274 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
67 fzofzp1 10183 . . . . . . . . . . . . . 14 (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
6867adantl 275 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
6966, 68ffvelrnd 5632 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
7054, 69sselid 3145 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ ℤ)
7152adantr 274 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
72 gcdcom 11928 . . . . . . . . . . 11 (((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = (𝑁 gcd (𝐹‘(𝑘 + 1))))
7370, 71, 72syl2anc 409 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = (𝑁 gcd (𝐹‘(𝑘 + 1))))
74 oveq1 5860 . . . . . . . . . . . . . 14 (𝑦 = (𝐹‘(𝑘 + 1)) → (𝑦 gcd 𝑁) = ((𝐹‘(𝑘 + 1)) gcd 𝑁))
7574eqeq1d 2179 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑘 + 1)) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
7675, 29elrab2 2889 . . . . . . . . . . . 12 ((𝐹‘(𝑘 + 1)) ∈ 𝑆 ↔ ((𝐹‘(𝑘 + 1)) ∈ (0..^𝑁) ∧ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
7776simprbi 273 . . . . . . . . . . 11 ((𝐹‘(𝑘 + 1)) ∈ 𝑆 → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1)
7869, 77syl 14 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1)
7973, 78eqtr3d 2205 . . . . . . . . 9 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1)
8079adantr 274 . . . . . . . 8 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1)
8128a1i 9 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 1 ∈ ℤ)
82 elfzoelz 10103 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ ℤ)
8382adantl 275 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ ℤ)
8481, 83fzfigd 10387 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (1...𝑘) ∈ Fin)
8538ad2antrr 485 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
86 elfznn 10010 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
8786nnred 8891 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℝ)
8887adantl 275 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ ℝ)
893nnred 8891 . . . . . . . . . . . . . . . 16 (𝜑 → (ϕ‘𝑁) ∈ ℝ)
9089ad2antrr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℝ)
9182ad2antlr 486 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℤ)
9291zred 9334 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℝ)
93 elfzle2 9984 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1...𝑘) → 𝑥𝑘)
9493adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥𝑘)
95 elfzolt2 10112 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 < (ϕ‘𝑁))
9695ad2antlr 486 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 < (ϕ‘𝑁))
9788, 92, 90, 94, 96lelttrd 8044 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 < (ϕ‘𝑁))
9888, 90, 97ltled 8038 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ (ϕ‘𝑁))
99 elfzuz 9977 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ (ℤ‘1))
10042ad2antrr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℤ)
101 elfz5 9973 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℤ‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
10299, 100, 101syl2an2 589 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
10398, 102mpbird 166 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ (1...(ϕ‘𝑁)))
10485, 103ffvelrnd 5632 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹𝑥) ∈ 𝑆)
10554, 104sselid 3145 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹𝑥) ∈ ℤ)
10684, 105fprodzcl 11572 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...𝑘)(𝐹𝑥) ∈ ℤ)
107 rpmul 12052 . . . . . . . . . 10 ((𝑁 ∈ ℤ ∧ ∏𝑥 ∈ (1...𝑘)(𝐹𝑥) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ∈ ℤ) → (((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1 ∧ (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
10871, 106, 70, 107syl3anc 1233 . . . . . . . . 9 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1 ∧ (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
109108adantr 274 . . . . . . . 8 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1 ∧ (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
11065, 80, 109mp2and 431 . . . . . . 7 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1)
111 elfzouz 10107 . . . . . . . . . . . 12 (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ (ℤ‘1))
112111adantl 275 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ (ℤ‘1))
11338ad2antrr 485 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
114 elfzelz 9981 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℤ)
115114zred 9334 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℝ)
116115adantl 275 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ ℝ)
11782ad2antlr 486 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑘 ∈ ℤ)
118117peano2zd 9337 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
119118zred 9334 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
12089ad2antrr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℝ)
121 elfzle2 9984 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ≤ (𝑘 + 1))
122121adantl 275 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (𝑘 + 1))
123 elfzle2 9984 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (1...(ϕ‘𝑁)) → (𝑘 + 1) ≤ (ϕ‘𝑁))
12467, 123syl 14 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ≤ (ϕ‘𝑁))
125124ad2antlr 486 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ≤ (ϕ‘𝑁))
126116, 119, 120, 122, 125letrd 8043 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (ϕ‘𝑁))
127 elfzuz 9977 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ (ℤ‘1))
12842ad2antrr 485 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℤ)
129127, 128, 101syl2an2 589 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
130126, 129mpbird 166 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ (1...(ϕ‘𝑁)))
131113, 130ffvelrnd 5632 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹𝑥) ∈ 𝑆)
13235, 131sselid 3145 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹𝑥) ∈ ℂ)
133 fveq2 5496 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
134112, 132, 133fprodp1 11563 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥) = (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1))))
135134oveq2d 5869 . . . . . . . . 9 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))))
136135eqeq1d 2179 . . . . . . . 8 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1 ↔ (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
137136adantr 274 . . . . . . 7 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → ((𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1 ↔ (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
138110, 137mpbird 166 . . . . . 6 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1)
139138ex 114 . . . . 5 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1))
140139expcom 115 . . . 4 (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝜑 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1)))
141140a2d 26 . . 3 (𝑘 ∈ (1..^(ϕ‘𝑁)) → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = 1)))
14212, 17, 22, 27, 64, 141fzind2 10195 . 2 ((ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)) → (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)) = 1))
1437, 142mpcom 36 1 (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)) = 1)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  {crab 2452   class class class wbr 3989  wf 5194  1-1-ontowf1o 5197  cfv 5198  (class class class)co 5853  cc 7772  cr 7773  0cc0 7774  1c1 7775   + caddc 7777   · cmul 7779   < clt 7954  cle 7955  cn 8878  0cn0 9135  cz 9212  cuz 9487  ...cfz 9965  ..^cfzo 10098  cprod 11513   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-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-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-proddc 11514  df-dvds 11750  df-gcd 11898  df-phi 12165
This theorem is referenced by:  eulerthlemth  12186
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