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Theorem eulerthlemrprm 12161

Description: Lemma for eulerth 12165. 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 999	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	phicld 12150	. . . 4 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
4		elnnuz 9502	. . . 4 ⊢ ((ϕ‘𝑁) ∈ ℕ ↔ (ϕ‘𝑁) ∈ (ℤ_≥‘1))
5	3, 4	sylib 121	. . 3 ⊢ (𝜑 → (ϕ‘𝑁) ∈ (ℤ_≥‘1))
6		eluzfz2 9967	. . 3 ⊢ ((ϕ‘𝑁) ∈ (ℤ_≥‘1) → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
8		oveq2 5850	. . . . . . 7 ⊢ (𝑤 = 1 → (1...𝑤) = (1...1))
9	8	prodeq1d 11505	. . . . . 6 ⊢ (𝑤 = 1 → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...1)(𝐹‘𝑥))
10	9	oveq2d 5858	. . . . 5 ⊢ (𝑤 = 1 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)))
11	10	eqeq1d 2174	. . . 4 ⊢ (𝑤 = 1 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = 1))
12	11	imbi2d 229	. . 3 ⊢ (𝑤 = 1 → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = 1)))
13		oveq2 5850	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (1...𝑤) = (1...𝑘))
14	13	prodeq1d 11505	. . . . . 6 ⊢ (𝑤 = 𝑘 → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥))
15	14	oveq2d 5858	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)))
16	15	eqeq1d 2174	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1))
17	16	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1) ↔ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)) = 1)))
18		oveq2 5850	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (1...𝑤) = (1...(𝑘 + 1)))
19	18	prodeq1d 11505	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥))
20	19	oveq2d 5858	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)))
21	20	eqeq1d 2174	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)) = 1))
22	21	imbi2d 229	. . 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 5850	. . . . . . 7 ⊢ (𝑤 = (ϕ‘𝑁) → (1...𝑤) = (1...(ϕ‘𝑁)))
24	23	prodeq1d 11505	. . . . . 6 ⊢ (𝑤 = (ϕ‘𝑁) → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))
25	24	oveq2d 5858	. . . . 5 ⊢ (𝑤 = (ϕ‘𝑁) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)))
26	25	eqeq1d 2174	. . . 4 ⊢ (𝑤 = (ϕ‘𝑁) → ((𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = 1 ↔ (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1))
27	26	imbi2d 229	. . 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 9217	. . . . . . 7 ⊢ 1 ∈ ℤ
29		eulerth.2	. . . . . . . . . . 11 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
30		ssrab2 3227	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
31	29, 30	eqsstri 3174	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (0..^𝑁)
32		fzo0ssnn0 10150	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℕ₀
33	31, 32	sstri 3151	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℕ₀
34		nn0sscn 9119	. . . . . . . . 9 ⊢ ℕ₀ ⊆ ℂ
35	33, 34	sstri 3151	. . . . . . . 8 ⊢ 𝑆 ⊆ ℂ
36		eulerth.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
37		f1of 5432	. . . . . . . . . 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 8900	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑁))
40		uzid 9480	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
41	28, 40	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ (ℤ_≥‘1)
42	3	nnzd 9312	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
43		elfz5 9952	. . . . . . . . . . 11 ⊢ ((1 ∈ (ℤ_≥‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
44	41, 42, 43	sylancr 411	. . . . . . . . . 10 ⊢ (𝜑 → (1 ∈ (1...(ϕ‘𝑁)) ↔ 1 ≤ (ϕ‘𝑁)))
45	39, 44	mpbird 166	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ (1...(ϕ‘𝑁)))
46	38, 45	ffvelrnd 5621	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑆)
47	35, 46	sselid 3140	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
48		fveq2 5486	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1))
49	48	fprod1 11535	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝐹‘1) ∈ ℂ) → ∏𝑥 ∈ (1...1)(𝐹‘𝑥) = (𝐹‘1))
50	28, 47, 49	sylancr 411	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...1)(𝐹‘𝑥) = (𝐹‘1))
51	50	oveq2d 5858	. . . . 5 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = (𝑁 gcd (𝐹‘1)))
52	2	nnzd 9312	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
53		nn0ssz 9209	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℤ
54	33, 53	sstri 3151	. . . . . . 7 ⊢ 𝑆 ⊆ ℤ
55	54, 46	sselid 3140	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℤ)
56		gcdcom 11906	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (𝐹‘1) ∈ ℤ) → (𝑁 gcd (𝐹‘1)) = ((𝐹‘1) gcd 𝑁))
57	52, 55, 56	syl2anc 409	. . . . 5 ⊢ (𝜑 → (𝑁 gcd (𝐹‘1)) = ((𝐹‘1) gcd 𝑁))
58		oveq1 5849	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘1) → (𝑦 gcd 𝑁) = ((𝐹‘1) gcd 𝑁))
59	58	eqeq1d 2174	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘1) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘1) gcd 𝑁) = 1))
60	59, 29	elrab2 2885	. . . . . . 7 ⊢ ((𝐹‘1) ∈ 𝑆 ↔ ((𝐹‘1) ∈ (0..^𝑁) ∧ ((𝐹‘1) gcd 𝑁) = 1))
61	46, 60	sylib 121	. . . . . 6 ⊢ (𝜑 → ((𝐹‘1) ∈ (0..^𝑁) ∧ ((𝐹‘1) gcd 𝑁) = 1))
62	61	simprd 113	. . . . 5 ⊢ (𝜑 → ((𝐹‘1) gcd 𝑁) = 1)
63	51, 57, 62	3eqtrd 2202	. . . 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 109	. . . . . . . 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 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
67		fzofzp1 10162	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
68	67	adantl 275	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑘 + 1) ∈ (1...(ϕ‘𝑁)))
69	66, 68	ffvelrnd 5621	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
70	54, 69	sselid 3140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ ℤ)
71	52	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
72		gcdcom 11906	. . . . . . . . . . 11 ⊢ (((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = (𝑁 gcd (𝐹‘(𝑘 + 1))))
73	70, 71, 72	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ((𝐹‘(𝑘 + 1)) gcd 𝑁) = (𝑁 gcd (𝐹‘(𝑘 + 1))))
74		oveq1 5849	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → (𝑦 gcd 𝑁) = ((𝐹‘(𝑘 + 1)) gcd 𝑁))
75	74	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
76	75, 29	elrab2 2885	. . . . . . . . . . . 12 ⊢ ((𝐹‘(𝑘 + 1)) ∈ 𝑆 ↔ ((𝐹‘(𝑘 + 1)) ∈ (0..^𝑁) ∧ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
77	76	simprbi 273	. . . . . . . . . . 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 2200	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝑁 gcd (𝐹‘(𝑘 + 1))) = 1)
80	79	adantr 274	. . . . . . . 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 10082	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ ℤ)
83	82	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ ℤ)
84	81, 83	fzfigd 10366	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (1...𝑘) ∈ Fin)
85	38	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
86		elfznn 9989	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
87	86	nnred 8870	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℝ)
88	87	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ ℝ)
89	3	nnred 8870	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℝ)
90	89	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℝ)
91	82	ad2antlr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℤ)
92	91	zred 9313	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℝ)
93		elfzle2 9963	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ≤ 𝑘)
94	93	adantl 275	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ 𝑘)
95		elfzolt2 10091	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 < (ϕ‘𝑁))
96	95	ad2antlr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 < (ϕ‘𝑁))
97	88, 92, 90, 94, 96	lelttrd 8023	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 < (ϕ‘𝑁))
98	88, 90, 97	ltled 8017	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ (ϕ‘𝑁))
99		elfzuz 9956	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ (ℤ_≥‘1))
100	42	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (ϕ‘𝑁) ∈ ℤ)
101		elfz5 9952	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℤ_≥‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
102	99, 100, 101	syl2an2 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
103	98, 102	mpbird 166	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ (1...(ϕ‘𝑁)))
104	85, 103	ffvelrnd 5621	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹‘𝑥) ∈ 𝑆)
105	54, 104	sselid 3140	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹‘𝑥) ∈ ℤ)
106	84, 105	fprodzcl 11550	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) ∈ ℤ)
107		rpmul 12030	. . . . . . . . . 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 1228	. . . . . . . . 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 274	. . . . . . . 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 430	. . . . . . 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 10086	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ (ℤ_≥‘1))
112	111	adantl 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ (ℤ_≥‘1))
113	38	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
114		elfzelz 9960	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℤ)
115	114	zred 9313	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℝ)
116	115	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ ℝ)
117	82	ad2antlr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑘 ∈ ℤ)
118	117	peano2zd 9316	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
119	118	zred 9313	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
120	89	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℝ)
121		elfzle2 9963	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ≤ (𝑘 + 1))
122	121	adantl 275	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (𝑘 + 1))
123		elfzle2 9963	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ≤ (ϕ‘𝑁))
126	116, 119, 120, 122, 125	letrd 8022	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (ϕ‘𝑁))
127		elfzuz 9956	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ (ℤ_≥‘1))
128	42	ad2antrr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (ϕ‘𝑁) ∈ ℤ)
129	127, 128, 101	syl2an2 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
130	126, 129	mpbird 166	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ∈ (1...(ϕ‘𝑁)))
131	113, 130	ffvelrnd 5621	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹‘𝑥) ∈ 𝑆)
132	35, 131	sselid 3140	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹‘𝑥) ∈ ℂ)
133		fveq2 5486	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
134	112, 132, 133	fprodp1 11541	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥) = (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1))))
135	134	oveq2d 5858	. . . . . . . . 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 2174	. . . . . . . 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 274	. . . . . . 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 166	. . . . . 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 114	. . . . 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 115	. . . 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 10174	. 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 103 ↔ wb 104 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 {crab 2448 class class class wbr 3982 ⟶wf 5184 –1-1-onto→wf1o 5187 ‘cfv 5188 (class class class)co 5842 ℂcc 7751 ℝcr 7752 0cc0 7753 1c1 7754 + caddc 7756 · cmul 7758 < clt 7933 ≤ cle 7934 ℕcn 8857 ℕ₀cn0 9114 ℤcz 9191 ℤ_≥cuz 9466 ...cfz 9944 ..^cfzo 10077 ∏cprod 11491 gcd cgcd 11875 ϕcphi 12141

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-fz 9945 df-fzo 10078 df-fl 10205 df-mod 10258 df-seqfrec 10381 df-exp 10455 df-ihash 10689 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-proddc 11492 df-dvds 11728 df-gcd 11876 df-phi 12143

This theorem is referenced by: eulerthlemth 12164

W3C validator