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Theorem eulerthlemrprm 12108
Description: Lemma for eulerth 12112. 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 994 . . . . 5 (𝜑𝑁 ∈ ℕ)
32phicld 12097 . . . 4 (𝜑 → (ϕ‘𝑁) ∈ ℕ)
4 elnnuz 9476 . . . 4 ((ϕ‘𝑁) ∈ ℕ ↔ (ϕ‘𝑁) ∈ (ℤ‘1))
53, 4sylib 121 . . 3 (𝜑 → (ϕ‘𝑁) ∈ (ℤ‘1))
6 eluzfz2 9935 . . 3 ((ϕ‘𝑁) ∈ (ℤ‘1) → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
75, 6syl 14 . 2 (𝜑 → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
8 oveq2 5833 . . . . . . 7 (𝑤 = 1 → (1...𝑤) = (1...1))
98prodeq1d 11465 . . . . . 6 (𝑤 = 1 → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...1)(𝐹𝑥))
109oveq2d 5841 . . . . 5 (𝑤 = 1 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)))
1110eqeq1d 2166 . . . 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 5833 . . . . . . 7 (𝑤 = 𝑘 → (1...𝑤) = (1...𝑘))
1413prodeq1d 11465 . . . . . 6 (𝑤 = 𝑘 → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...𝑘)(𝐹𝑥))
1514oveq2d 5841 . . . . 5 (𝑤 = 𝑘 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)))
1615eqeq1d 2166 . . . 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 5833 . . . . . . 7 (𝑤 = (𝑘 + 1) → (1...𝑤) = (1...(𝑘 + 1)))
1918prodeq1d 11465 . . . . . 6 (𝑤 = (𝑘 + 1) → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥))
2019oveq2d 5841 . . . . 5 (𝑤 = (𝑘 + 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)))
2120eqeq1d 2166 . . . 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 5833 . . . . . . 7 (𝑤 = (ϕ‘𝑁) → (1...𝑤) = (1...(ϕ‘𝑁)))
2423prodeq1d 11465 . . . . . 6 (𝑤 = (ϕ‘𝑁) → ∏𝑥 ∈ (1...𝑤)(𝐹𝑥) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥))
2524oveq2d 5841 . . . . 5 (𝑤 = (ϕ‘𝑁) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹𝑥)))
2625eqeq1d 2166 . . . 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 9194 . . . . . . 7 1 ∈ ℤ
29 eulerth.2 . . . . . . . . . . 11 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
30 ssrab2 3213 . . . . . . . . . . 11 {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
3129, 30eqsstri 3160 . . . . . . . . . 10 𝑆 ⊆ (0..^𝑁)
32 fzo0ssnn0 10118 . . . . . . . . . 10 (0..^𝑁) ⊆ ℕ0
3331, 32sstri 3137 . . . . . . . . 9 𝑆 ⊆ ℕ0
34 nn0sscn 9096 . . . . . . . . 9 0 ⊆ ℂ
3533, 34sstri 3137 . . . . . . . 8 𝑆 ⊆ ℂ
36 eulerth.4 . . . . . . . . . 10 (𝜑𝐹:(1...(ϕ‘𝑁))–1-1-onto𝑆)
37 f1of 5415 . . . . . . . . . 10 (𝐹:(1...(ϕ‘𝑁))–1-1-onto𝑆𝐹:(1...(ϕ‘𝑁))⟶𝑆)
3836, 37syl 14 . . . . . . . . 9 (𝜑𝐹:(1...(ϕ‘𝑁))⟶𝑆)
393nnge1d 8877 . . . . . . . . . 10 (𝜑 → 1 ≤ (ϕ‘𝑁))
40 uzid 9454 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
4128, 40ax-mp 5 . . . . . . . . . . 11 1 ∈ (ℤ‘1)
423nnzd 9286 . . . . . . . . . . 11 (𝜑 → (ϕ‘𝑁) ∈ ℤ)
43 elfz5 9921 . . . . . . . . . . 11 ((1 ∈ (ℤ‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
4441, 42, 43sylancr 411 . . . . . . . . . 10 (𝜑 → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
4539, 44mpbird 166 . . . . . . . . 9 (𝜑 → 1 ∈ (1...(ϕ‘𝑁)))
4638, 45ffvelrnd 5604 . . . . . . . 8 (𝜑 → (𝐹‘1) ∈ 𝑆)
4735, 46sseldi 3126 . . . . . . 7 (𝜑 → (𝐹‘1) ∈ ℂ)
48 fveq2 5469 . . . . . . . 8 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
4948fprod1 11495 . . . . . . 7 ((1 ∈ ℤ ∧ (𝐹‘1) ∈ ℂ) → ∏𝑥 ∈ (1...1)(𝐹𝑥) = (𝐹‘1))
5028, 47, 49sylancr 411 . . . . . 6 (𝜑 → ∏𝑥 ∈ (1...1)(𝐹𝑥) = (𝐹‘1))
5150oveq2d 5841 . . . . 5 (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹𝑥)) = (𝑁 gcd (𝐹‘1)))
522nnzd 9286 . . . . . 6 (𝜑𝑁 ∈ ℤ)
53 nn0ssz 9186 . . . . . . . 8 0 ⊆ ℤ
5433, 53sstri 3137 . . . . . . 7 𝑆 ⊆ ℤ
5554, 46sseldi 3126 . . . . . 6 (𝜑 → (𝐹‘1) ∈ ℤ)
56 gcdcom 11861 . . . . . 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 5832 . . . . . . . . 9 (𝑦 = (𝐹‘1) → (𝑦 gcd 𝑁) = ((𝐹‘1) gcd 𝑁))
5958eqeq1d 2166 . . . . . . . 8 (𝑦 = (𝐹‘1) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘1) gcd 𝑁) = 1))
6059, 29elrab2 2871 . . . . . . 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 2194 . . . 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 10130 . . . . . . . . . . . . . 14 (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
6867adantl 275 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
6966, 68ffvelrnd 5604 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
7054, 69sseldi 3126 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ ℤ)
7152adantr 274 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
72 gcdcom 11861 . . . . . . . . . . 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 5832 . . . . . . . . . . . . . 14 (𝑦 = (𝐹‘(𝑘 + 1)) → (𝑦 gcd 𝑁) = ((𝐹‘(𝑘 + 1)) gcd 𝑁))
7574eqeq1d 2166 . . . . . . . . . . . . 13 (𝑦 = (𝐹‘(𝑘 + 1)) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
7675, 29elrab2 2871 . . . . . . . . . . . 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 2192 . . . . . . . . 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 10050 . . . . . . . . . . . . 13 (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ ℤ)
8382adantl 275 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ ℤ)
8481, 83fzfigd 10334 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (1...𝑘) ∈ Fin)
8538ad2antrr 480 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
86 elfznn 9957 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
8786nnred 8847 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℝ)
8887adantl 275 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ ℝ)
893nnred 8847 . . . . . . . . . . . . . . . 16 (𝜑 → (ϕ‘𝑁) ∈ ℝ)
9089ad2antrr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℝ)
9182ad2antlr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℤ)
9291zred 9287 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℝ)
93 elfzle2 9931 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1...𝑘) → 𝑥𝑘)
9493adantl 275 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥𝑘)
95 elfzolt2 10059 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 < (ϕ‘𝑁))
9695ad2antlr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 < (ϕ‘𝑁))
9788, 92, 90, 94, 96lelttrd 8001 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 < (ϕ‘𝑁))
9888, 90, 97ltled 7995 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ (ϕ‘𝑁))
99 elfzuz 9925 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1...𝑘) → 𝑥 ∈ (ℤ‘1))
10042ad2antrr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℤ)
101 elfz5 9921 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (ℤ‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
10299, 100, 101syl2an2 584 . . . . . . . . . . . . . 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 5604 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹𝑥) ∈ 𝑆)
10554, 104sseldi 3126 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹𝑥) ∈ ℤ)
10684, 105fprodzcl 11510 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...𝑘)(𝐹𝑥) ∈ ℤ)
107 rpmul 11979 . . . . . . . . . 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 1220 . . . . . . . . 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 430 . . . . . . 7 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹𝑥)) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))) = 1)
111 elfzouz 10054 . . . . . . . . . . . 12 (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ (ℤ‘1))
112111adantl 275 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ (ℤ‘1))
11338ad2antrr 480 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
114 elfzelz 9929 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℤ)
115114zred 9287 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℝ)
116115adantl 275 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ ℝ)
11782ad2antlr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑘 ∈ ℤ)
118117peano2zd 9290 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
119118zred 9287 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
12089ad2antrr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℝ)
121 elfzle2 9931 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ≤ (𝑘 + 1))
122121adantl 275 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (𝑘 + 1))
123 elfzle2 9931 . . . . . . . . . . . . . . . . 17 ((𝑘 + 1) ∈ (1...(ϕ‘𝑁)) → (𝑘 + 1) ≤ (ϕ‘𝑁))
12467, 123syl 14 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ≤ (ϕ‘𝑁))
125124ad2antlr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ≤ (ϕ‘𝑁))
126116, 119, 120, 122, 125letrd 8000 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (ϕ‘𝑁))
127 elfzuz 9925 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ (ℤ‘1))
12842ad2antrr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℤ)
129127, 128, 101syl2an2 584 . . . . . . . . . . . . . 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 5604 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹𝑥) ∈ 𝑆)
13235, 131sseldi 3126 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹𝑥) ∈ ℂ)
133 fveq2 5469 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → (𝐹𝑥) = (𝐹‘(𝑘 + 1)))
134112, 132, 133fprodp1 11501 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥) = (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1))))
135134oveq2d 5841 . . . . . . . . 9 ((𝜑𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹𝑥)) = (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹𝑥) · (𝐹‘(𝑘 + 1)))))
136135eqeq1d 2166 . . . . . . . 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 10142 . 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 963   = wceq 1335  wcel 2128  {crab 2439   class class class wbr 3966  wf 5167  1-1-ontowf1o 5170  cfv 5171  (class class class)co 5825  cc 7731  cr 7732  0cc0 7733  1c1 7734   + caddc 7736   · cmul 7738   < clt 7913  cle 7914  cn 8834  0cn0 9091  cz 9168  cuz 9440  ...cfz 9913  ..^cfzo 10045  cprod 11451   gcd cgcd 11833  ϕcphi 12088
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 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853
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 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-frec 6339  df-1o 6364  df-oadd 6368  df-er 6481  df-en 6687  df-dom 6688  df-fin 6689  df-sup 6929  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-fz 9914  df-fzo 10046  df-fl 10173  df-mod 10226  df-seqfrec 10349  df-exp 10423  df-ihash 10654  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-clim 11180  df-proddc 11452  df-dvds 11688  df-gcd 11834  df-phi 12090
This theorem is referenced by:  eulerthlemth  12111
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