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Theorem eulerthlemrprm 12229

Description: Lemma for eulerth 12233. 𝑁 and ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) 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	⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)

**Assertion**
Ref	Expression
eulerthlemrprm	⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 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.

Step	Hyp	Ref	Expression
1		eulerth.1	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
2	1	simp1d 1009	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	phicld 12218	. . . 4 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
4		elnnuz 9564	. . . 4 ⊢ ((ϕ‘𝑁) ∈ ℕ ↔ (ϕ‘𝑁) ∈ (ℤ_≥‘1))
5	3, 4	sylib 122	. . 3 ⊢ (𝜑 → (ϕ‘𝑁) ∈ (ℤ_≥‘1))
6		eluzfz2 10032	. . 3 ⊢ ((ϕ‘𝑁) ∈ (ℤ_≥‘1) → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
8		oveq2 5883	. . . . . . 7 ⊢ (𝑤 = 1 → (1...𝑤) = (1...1))
9	8	prodeq1d 11572	. . . . . 6 ⊢ (𝑤 = 1 → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...1)(𝐹‘𝑥))
10	9	oveq2d 5891	. . . . 5 ⊢ (𝑤 = 1 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)))
11	10	eqeq1d 2186	. . . 4 ⊢ (𝑤 = 1 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = 1))
12	11	imbi2d 230	. . 3 ⊢ (𝑤 = 1 → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = 1)))
13		oveq2 5883	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (1...𝑤) = (1...𝑘))
14	13	prodeq1d 11572	. . . . . 6 ⊢ (𝑤 = 𝑘 → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥))
15	14	oveq2d 5891	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)))
16	15	eqeq1d 2186	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1))
17	16	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1)))
18		oveq2 5883	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (1...𝑤) = (1...(𝑘 + 1)))
19	18	prodeq1d 11572	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥))
20	19	oveq2d 5891	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)))
21	20	eqeq1d 2186	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1))
22	21	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1)))
23		oveq2 5883	. . . . . . 7 ⊢ (𝑤 = (ϕ‘𝑁) → (1...𝑤) = (1...(ϕ‘𝑁)))
24	23	prodeq1d 11572	. . . . . 6 ⊢ (𝑤 = (ϕ‘𝑁) → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))
25	24	oveq2d 5891	. . . . 5 ⊢ (𝑤 = (ϕ‘𝑁) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)))
26	25	eqeq1d 2186	. . . 4 ⊢ (𝑤 = (ϕ‘𝑁) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1))
27	26	imbi2d 230	. . 3 ⊢ (𝑤 = (ϕ‘𝑁) → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)))
28		1z 9279	. . . . . . 7 ⊢ 1 ∈ ℤ
29		eulerth.2	. . . . . . . . . . 11 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
30		ssrab2 3241	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
31	29, 30	eqsstri 3188	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (0..^𝑁)
32		fzo0ssnn0 10215	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℕ₀
33	31, 32	sstri 3165	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℕ₀
34		nn0sscn 9181	. . . . . . . . 9 ⊢ ℕ₀ ⊆ ℂ
35	33, 34	sstri 3165	. . . . . . . 8 ⊢ 𝑆 ⊆ ℂ
36		eulerth.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
37		f1of 5462	. . . . . . . . . 10 ⊢ (𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆 → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
38	36, 37	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
39	3	nnge1d 8962	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑁))
40		uzid 9542	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
41	28, 40	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ (ℤ_≥‘1)
42	3	nnzd 9374	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
43		elfz5 10017	. . . . . . . . . . 11 ⊢ ((1 ∈ (ℤ_≥‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
44	41, 42, 43	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
45	39, 44	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ (1...(ϕ‘𝑁)))
46	38, 45	ffvelcdmd 5653	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑆)
47	35, 46	sselid 3154	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
48		fveq2 5516	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1))
49	48	fprod1 11602	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝐹‘1) ∈ ℂ) → ∏𝑥 ∈ (1...1)(𝐹‘𝑥) = (𝐹‘1))
50	28, 47, 49	sylancr 414	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...1)(𝐹‘𝑥) = (𝐹‘1))
51	50	oveq2d 5891	. . . . 5 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = (𝑁 gcd (𝐹‘1)))
52	2	nnzd 9374	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
53		nn0ssz 9271	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℤ
54	33, 53	sstri 3165	. . . . . . 7 ⊢ 𝑆 ⊆ ℤ
55	54, 46	sselid 3154	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℤ)
56		gcdcom 11974	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (𝐹‘1) ∈ ℤ) → (𝑁 gcd (𝐹‘1)) = ((𝐹‘1) gcd 𝑁))
57	52, 55, 56	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑁 gcd (𝐹‘1)) = ((𝐹‘1) gcd 𝑁))
58		oveq1 5882	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘1) → (𝑦 gcd 𝑁) = ((𝐹‘1) gcd 𝑁))
59	58	eqeq1d 2186	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘1) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘1) gcd 𝑁) = 1))
60	59, 29	elrab2 2897	. . . . . . 7 ⊢ ((𝐹‘1) ∈ 𝑆 ↔ ((𝐹‘1) ∈ (0..^𝑁) ∧ ((𝐹‘1) gcd 𝑁) = 1))
61	46, 60	sylib 122	. . . . . 6 ⊢ (𝜑 → ((𝐹‘1) ∈ (0..^𝑁) ∧ ((𝐹‘1) gcd 𝑁) = 1))
62	61	simprd 114	. . . . 5 ⊢ (𝜑 → ((𝐹‘1) gcd 𝑁) = 1)
63	51, 57, 62	3eqtrd 2214	. . . 4 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = 1)
64	63	a1i 9	. . 3 ⊢ ((ϕ‘𝑁) ∈ (ℤ_≥‘1) → (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = 1))
65		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1)
66	38	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
67		fzofzp1 10227	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
68	67	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
69	66, 68	ffvelcdmd 5653	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
70	54, 69	sselid 3154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ ℤ)
71	52	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
72		gcdcom 11974	. . . . . . . . . . 11 ⊢ (((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = (𝑁 gcd (𝐹‘(𝑘 + 1))))
73	70, 71, 72	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = (𝑁 gcd (𝐹‘(𝑘 + 1))))
74		oveq1 5882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → (𝑦 gcd 𝑁) = ((𝐹‘(𝑘 + 1)) gcd 𝑁))
75	74	eqeq1d 2186	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
76	75, 29	elrab2 2897	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑘 + 1)) ∈ 𝑆 ↔ ((𝐹‘(𝑘 + 1)) ∈ (0..^𝑁) ∧ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
77	76	simprbi 275	. . . . . . . . . . 11 ⊢ ((𝐹‘(𝑘 + 1)) ∈ 𝑆 → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1)
78	69, 77	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1)
79	73, 78	eqtr3d 2212	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1)
80	79	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1)
81	28	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 1 ∈ ℤ)
82		elfzoelz 10147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ ℤ)
83	82	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ ℤ)
84	81, 83	fzfigd 10431	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (1...𝑘) ∈ Fin)
85	38	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
86		elfznn 10054	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
87	86	nnred 8932	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℝ)
88	87	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ ℝ)
89	3	nnred 8932	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℝ)
90	89	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℝ)
91	82	ad2antlr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℤ)
92	91	zred 9375	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℝ)
93		elfzle2 10028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ≤ 𝑘)
94	93	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ 𝑘)
95		elfzolt2 10156	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 < (ϕ‘𝑁))
96	95	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 < (ϕ‘𝑁))
97	88, 92, 90, 94, 96	lelttrd 8082	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 < (ϕ‘𝑁))
98	88, 90, 97	ltled 8076	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ (ϕ‘𝑁))
99		elfzuz 10021	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ (ℤ_≥‘1))
100	42	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℤ)
101		elfz5 10017	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℤ_≥‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
102	99, 100, 101	syl2an2 594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
103	98, 102	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ (1...(ϕ‘𝑁)))
104	85, 103	ffvelcdmd 5653	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹‘𝑥) ∈ 𝑆)
105	54, 104	sselid 3154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹‘𝑥) ∈ ℤ)
106	84, 105	fprodzcl 11617	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) ∈ ℤ)
107		rpmul 12098	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) ∈ ℤ ∧ (𝐹‘(𝑘 + 1)) ∈ ℤ) → (((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1 ∧ (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
108	71, 106, 70, 107	syl3anc 1238	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1 ∧ (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
109	108	adantr 276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → (((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1 ∧ (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
110	65, 80, 109	mp2and 433	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))) = 1)
111		elfzouz 10151	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ (ℤ_≥‘1))
112	111	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ (ℤ_≥‘1))
113	38	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
114		elfzelz 10025	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℤ)
115	114	zred 9375	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℝ)
116	115	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ ℝ)
117	82	ad2antlr 489	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑘 ∈ ℤ)
118	117	peano2zd 9378	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
119	118	zred 9375	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
120	89	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℝ)
121		elfzle2 10028	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ≤ (𝑘 + 1))
122	121	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (𝑘 + 1))
123		elfzle2 10028	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ (1...(ϕ‘𝑁)) → (𝑘 + 1) ≤ (ϕ‘𝑁))
124	67, 123	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ≤ (ϕ‘𝑁))
125	124	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ≤ (ϕ‘𝑁))
126	116, 119, 120, 122, 125	letrd 8081	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (ϕ‘𝑁))
127		elfzuz 10021	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ (ℤ_≥‘1))
128	42	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℤ)
129	127, 128, 101	syl2an2 594	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
130	126, 129	mpbird 167	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ (1...(ϕ‘𝑁)))
131	113, 130	ffvelcdmd 5653	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹‘𝑥) ∈ 𝑆)
132	35, 131	sselid 3154	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹‘𝑥) ∈ ℂ)
133		fveq2 5516	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
134	112, 132, 133	fprodp1 11608	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥) = (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1))))
135	134	oveq2d 5891	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))))
136	135	eqeq1d 2186	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
137	136	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → ((𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1)))) = 1))
138	110, 137	mpbird 167	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1)
139	138	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1))
140	139	expcom 116	. . . 4 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝜑 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1)))
141	140	a2d 26	. . 3 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1) → (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1)))
142	12, 17, 22, 27, 64, 141	fzind2 10239	. 2 ⊢ ((ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)) → (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1))
143	7, 142	mpcom 36	1 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {crab 2459 class class class wbr 4004 ⟶wf 5213 –1-1-onto→wf1o 5216 ‘cfv 5217 (class class class)co 5875 ℂcc 7809 ℝcr 7810 0cc0 7811 1c1 7812 + caddc 7814 · cmul 7816 < clt 7992 ≤ cle 7993 ℕcn 8919 ℕ₀cn0 9176 ℤcz 9253 ℤ_≥cuz 9528 ...cfz 10008 ..^cfzo 10142 ∏cprod 11558 gcd cgcd 11943 ϕcphi 12209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-oadd 6421 df-er 6535 df-en 6741 df-dom 6742 df-fin 6743 df-sup 6983 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-fl 10270 df-mod 10323 df-seqfrec 10446 df-exp 10520 df-ihash 10756 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-proddc 11559 df-dvds 11795 df-gcd 11944 df-phi 12211

This theorem is referenced by: eulerthlemth 12232

W3C validator