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Theorem eulerthlemrprm 12934

Description: Lemma for eulerth 12938. 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 1036	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	phicld 12923	. . . 4 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
4		elnnuz 9897	. . . 4 ⊢ ((ϕ‘𝑁) ∈ ℕ ↔ (ϕ‘𝑁) ∈ (ℤ_≥‘1))
5	3, 4	sylib 122	. . 3 ⊢ (𝜑 → (ϕ‘𝑁) ∈ (ℤ_≥‘1))
6		eluzfz2 10372	. . 3 ⊢ ((ϕ‘𝑁) ∈ (ℤ_≥‘1) → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
7	5, 6	syl 14	. 2 ⊢ (𝜑 → (ϕ‘𝑁) ∈ (1...(ϕ‘𝑁)))
8		oveq2 6060	. . . . . . 7 ⊢ (𝑤 = 1 → (1...𝑤) = (1...1))
9	8	prodeq1d 12258	. . . . . 6 ⊢ (𝑤 = 1 → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...1)(𝐹‘𝑥))
10	9	oveq2d 6068	. . . . 5 ⊢ (𝑤 = 1 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)))
11	10	eqeq1d 2243	. . . 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 6060	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (1...𝑤) = (1...𝑘))
14	13	prodeq1d 12258	. . . . . 6 ⊢ (𝑤 = 𝑘 → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥))
15	14	oveq2d 6068	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥)))
16	15	eqeq1d 2243	. . . 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 6060	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (1...𝑤) = (1...(𝑘 + 1)))
19	18	prodeq1d 12258	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥))
20	19	oveq2d 6068	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥)))
21	20	eqeq1d 2243	. . . 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 6060	. . . . . . 7 ⊢ (𝑤 = (ϕ‘𝑁) → (1...𝑤) = (1...(ϕ‘𝑁)))
24	23	prodeq1d 12258	. . . . . 6 ⊢ (𝑤 = (ϕ‘𝑁) → ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))
25	24	oveq2d 6068	. . . . 5 ⊢ (𝑤 = (ϕ‘𝑁) → (𝑁 gcd ∏𝑥 ∈ (1...𝑤)(𝐹‘𝑥)) = (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)))
26	25	eqeq1d 2243	. . . 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 9608	. . . . . . 7 ⊢ 1 ∈ ℤ
29		eulerth.2	. . . . . . . . . . 11 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
30		ssrab2 3325	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
31	29, 30	eqsstri 3272	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (0..^𝑁)
32		fzo0ssnn0 10567	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℕ₀
33	31, 32	sstri 3249	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℕ₀
34		nn0sscn 9506	. . . . . . . . 9 ⊢ ℕ₀ ⊆ ℂ
35	33, 34	sstri 3249	. . . . . . . 8 ⊢ 𝑆 ⊆ ℂ
36		eulerth.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
37		f1of 5616	. . . . . . . . . 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 9285	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑁))
40		uzid 9874	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
41	28, 40	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ (ℤ_≥‘1)
42	3	nnzd 9705	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
43		elfz5 10357	. . . . . . . . . . 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 5815	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘1) ∈ 𝑆)
47	35, 46	sselid 3238	. . . . . . 7 ⊢ (𝜑 → (𝐹‘1) ∈ ℂ)
48		fveq2 5672	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝐹‘𝑥) = (𝐹‘1))
49	48	fprod1 12288	. . . . . . 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 6068	. . . . 5 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...1)(𝐹‘𝑥)) = (𝑁 gcd (𝐹‘1)))
52	2	nnzd 9705	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
53		nn0ssz 9600	. . . . . . . 8 ⊢ ℕ₀ ⊆ ℤ
54	33, 53	sstri 3249	. . . . . . 7 ⊢ 𝑆 ⊆ ℤ
55	54, 46	sselid 3238	. . . . . 6 ⊢ (𝜑 → (𝐹‘1) ∈ ℤ)
56		gcdcom 12677	. . . . . 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 6059	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘1) → (𝑦 gcd 𝑁) = ((𝐹‘1) gcd 𝑁))
59	58	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘1) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘1) gcd 𝑁) = 1))
60	59, 29	elrab2 2978	. . . . . . 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 2271	. . . 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 10579	. . . . . . . . . . . . . 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 5815	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
70	54, 69	sselid 3238	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → (𝐹‘(𝑘 + 1)) ∈ ℤ)
71	52	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
72		gcdcom 12677	. . . . . . . . . . 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 6059	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → (𝑦 gcd 𝑁) = ((𝐹‘(𝑘 + 1)) gcd 𝑁))
75	74	eqeq1d 2243	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐹‘(𝑘 + 1)) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘(𝑘 + 1)) gcd 𝑁) = 1))
76	75, 29	elrab2 2978	. . . . . . . . . . . 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 2269	. . . . . . . . 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 10488	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1..^(ϕ‘𝑁)) → 𝑘 ∈ ℤ)
83	82	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → 𝑘 ∈ ℤ)
84	81, 83	fzfigd 10800	. . . . . . . . . . 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 10394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ)
87	86	nnred 9255	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℝ)
88	87	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ∈ ℝ)
89	3	nnred 9255	. . . . . . . . . . . . . . . 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 9706	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑘 ∈ ℝ)
93		elfzle2 10368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ≤ 𝑘)
94	93	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ 𝑘)
95		elfzolt2 10498	. . . . . . . . . . . . . . . . 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 8403	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 < (ϕ‘𝑁))
98	88, 90, 97	ltled 8397	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → 𝑥 ≤ (ϕ‘𝑁))
99		elfzuz 10361	. . . . . . . . . . . . . . 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 10357	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (ℤ_≥‘1) ∧ (ϕ‘𝑁) ∈ ℤ) → (𝑥 ∈ (1...(ϕ‘𝑁)) ↔ 𝑥 ≤ (ϕ‘𝑁)))
102	99, 100, 101	syl2an2 598	. . . . . . . . . . . . . 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 5815	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹‘𝑥) ∈ 𝑆)
105	54, 104	sselid 3238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...𝑘)) → (𝐹‘𝑥) ∈ ℤ)
106	84, 105	fprodzcl 12303	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) ∈ ℤ)
107		rpmul 12803	. . . . . . . . . 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 1274	. . . . . . . . 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 10492	. . . . . . . . . . . 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 10365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (1...(𝑘 + 1)) → 𝑥 ∈ ℤ)
115	114	zred 9706	. . . . . . . . . . . . . . . 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 9709	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
119	118	zred 9706	. . . . . . . . . . . . . . 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 10368	. . . . . . . . . . . . . . . 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 10368	. . . . . . . . . . . . . . . . 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 8402	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → 𝑥 ≤ (ϕ‘𝑁))
127		elfzuz 10361	. . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . 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 5815	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹‘𝑥) ∈ 𝑆)
132	35, 131	sselid 3238	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) ∧ 𝑥 ∈ (1...(𝑘 + 1))) → (𝐹‘𝑥) ∈ ℂ)
133		fveq2 5672	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
134	112, 132, 133	fprodp1 12294	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1..^(ϕ‘𝑁))) → ∏𝑥 ∈ (1...(𝑘 + 1))(𝐹‘𝑥) = (∏𝑥 ∈ (1...𝑘)(𝐹‘𝑥) · (𝐹‘(𝑘 + 1))))
135	134	oveq2d 6068	. . . . . . . . 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 2243	. . . . . . . 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 10592	. 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 1005 = wceq 1398 ∈ wcel 2205 {crab 2526 class class class wbr 4111 ⟶wf 5350 –1-1-onto→wf1o 5353 ‘cfv 5354 (class class class)co 6052 ℂcc 8130 ℝcr 8131 0cc0 8132 1c1 8133 + caddc 8135 · cmul 8137 < clt 8313 ≤ cle 8314 ℕcn 9242 ℕ₀cn0 9501 ℤcz 9582 ℤ_≥cuz 9859 ...cfz 10348 ..^cfzo 10483 ∏cprod 12244 gcd cgcd 12657 ϕcphi 12914

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250 ax-arch 8251 ax-caucvg 8252

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-oadd 6653 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-sup 7277 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-fz 10349 df-fzo 10484 df-fl 10637 df-mod 10692 df-seqfrec 10817 df-exp 10908 df-ihash 11147 df-cj 11535 df-re 11536 df-im 11537 df-rsqrt 11691 df-abs 11692 df-clim 11972 df-proddc 12245 df-dvds 12482 df-gcd 12658 df-phi 12916

This theorem is referenced by: eulerthlemth 12937

W3C validator