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Theorem eulerthlemth 12186

Description: Lemma for eulerth 12187. The result. (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
eulerthlemth	⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))

Distinct variable groups: 𝑦,𝐴 𝑦,𝐹 𝑦,𝑁 𝜑,𝑦 Allowed substitution hint: 𝑆(𝑦)

Proof of Theorem eulerthlemth Dummy variables 𝑢 𝑣 𝑥 𝑧 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		eulerth.2	. . . . . 6 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
3		eulerth.4	. . . . . 6 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
4	1, 2, 3	eulerthlema 12184	. . . . 5 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁))
5	1	simp1d 1004	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	1	simp2d 1005	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	5	phicld 12172	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
8	7	nnnn0d 9188	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
9		zexpcl 10491	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
10	6, 8, 9	syl2anc 409	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
11		1zzd 9239	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
12	7	nnzd 9333	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
13	11, 12	fzfigd 10387	. . . . . . . 8 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ∈ Fin)
14		ssrab2 3232	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
15	2, 14	eqsstri 3179	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (0..^𝑁)
16		fzo0ssnn0 10171	. . . . . . . . . . 11 ⊢ (0..^𝑁) ⊆ ℕ₀
17		nn0ssz 9230	. . . . . . . . . . 11 ⊢ ℕ₀ ⊆ ℤ
18	16, 17	sstri 3156	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℤ
19	15, 18	sstri 3156	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℤ
20		f1of 5442	. . . . . . . . . . 11 ⊢ (𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆 → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
21	3, 20	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
22	21	ffvelrnda 5631	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑥) ∈ 𝑆)
23	19, 22	sselid 3145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑥) ∈ ℤ)
24	13, 23	fprodzcl 11572	. . . . . . 7 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℤ)
25	10, 24	zmulcld 9340	. . . . . 6 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) ∈ ℤ)
26		fveq2 5496	. . . . . . . . 9 ⊢ (𝑧 = (^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁)) → (𝐹‘𝑧) = (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
27		eqid 2170	. . . . . . . . . 10 ⊢ (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))) = (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))
28	1, 2, 3, 27	eulerthlemh 12185	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
29		eqid 2170	. . . . . . . . . . . . 13 ⊢ (1...(ϕ‘𝑁)) = (1...(ϕ‘𝑁))
30		fveq2 5496	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
31	30	oveq2d 5869	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (𝐴 · (𝐹‘𝑣)) = (𝐴 · (𝐹‘𝑢)))
32	31	oveq1d 5868	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
33	32	cbvmptv 4085	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)) = (𝑢 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
34	1, 2, 29, 3, 33	eulerthlem1 12181	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
35		fveq2 5496	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
36	35	oveq2d 5869	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝐴 · (𝐹‘𝑣)) = (𝐴 · (𝐹‘𝑦)))
37	36	oveq1d 5868	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) = ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
38	37	cbvmptv 4085	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
39	38	feq1i 5340	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆 ↔ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
40	34, 39	sylib 121	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
41		fvco3 5567	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))‘𝑥) = (^◡𝐹‘((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑥)))
42	40, 41	sylan 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))‘𝑥) = (^◡𝐹‘((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑥)))
43		eqid 2170	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
44		fveq2 5496	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
45	44	oveq2d 5869	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑥)))
46	45	oveq1d 5868	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐴 · (𝐹‘𝑦)) mod 𝑁) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
47		simpr 109	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑥 ∈ (1...(ϕ‘𝑁)))
48	6	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝐴 ∈ ℤ)
49	48, 23	zmulcld 9340	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐴 · (𝐹‘𝑥)) ∈ ℤ)
50	5	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑁 ∈ ℕ)
51		zmodfzo 10303	. . . . . . . . . . . . 13 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
52	49, 50, 51	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
53	43, 46, 47, 52	fvmptd3 5589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑥) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
54	53	fveq2d 5500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (^◡𝐹‘((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑥)) = (^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁)))
55	42, 54	eqtrd 2203	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))‘𝑥) = (^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁)))
56	21	ffvelrnda 5631	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ 𝑆)
57	19, 56	sselid 3145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ ℤ)
58	57	zcnd 9335	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ ℂ)
59	26, 13, 28, 55, 58	fprodf1o 11551	. . . . . . . 8 ⊢ (𝜑 → ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
60	3	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
61		modgcd 11946	. . . . . . . . . . . . 13 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
62	49, 50, 61	syl2anc 409	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
63	50	nnzd 9333	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
64	63, 49	gcdcomd 11929	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
65	5	nnzd 9333	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
66	6, 65	gcdcomd 11929	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴))
67	1	simp3d 1006	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
68	66, 67	eqtr3d 2205	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
69	68	adantr 274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd 𝐴) = 1)
70	23, 63	gcdcomd 11929	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) gcd 𝑁) = (𝑁 gcd (𝐹‘𝑥)))
71		oveq1 5860	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 gcd 𝑁) = ((𝐹‘𝑥) gcd 𝑁))
72	71	eqeq1d 2179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘𝑥) gcd 𝑁) = 1))
73	72, 2	elrab2 2889	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ∈ 𝑆 ↔ ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
74	22, 73	sylib 121	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
75	74	simprd 113	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) gcd 𝑁) = 1)
76	70, 75	eqtr3d 2205	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐹‘𝑥)) = 1)
77		rpmul 12052	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐹‘𝑥) ∈ ℤ) → (((𝑁 gcd 𝐴) = 1 ∧ (𝑁 gcd (𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1))
78	63, 48, 23, 77	syl3anc 1233	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝑁 gcd 𝐴) = 1 ∧ (𝑁 gcd (𝐹‘𝑥)) = 1) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1))
79	69, 76, 78	mp2and 431	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1)
80	62, 64, 79	3eqtr2d 2209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1)
81		oveq1 5860	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → (𝑦 gcd 𝑁) = (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁))
82	81	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
83	82, 2	elrab2 2889	. . . . . . . . . . 11 ⊢ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁) ∧ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
84	52, 80, 83	sylanbrc 415	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆)
85		f1ocnvfv2 5757	. . . . . . . . . 10 ⊢ ((𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆 ∧ ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆) → (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
86	60, 84, 85	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
87	86	prodeq2dv 11529	. . . . . . . 8 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))) = ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁))
88	59, 87	eqtr2d 2204	. . . . . . 7 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) = ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧))
89		fveq2 5496	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
90	89	cbvprodv 11522	. . . . . . . 8 ⊢ ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)
91	90, 24	eqeltrid 2257	. . . . . . 7 ⊢ (𝜑 → ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) ∈ ℤ)
92	88, 91	eqeltrd 2247	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ ℤ)
93		moddvds 11761	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) ∈ ℤ ∧ ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ ℤ) → ((((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁) ↔ 𝑁 ∥ (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
94	5, 25, 92, 93	syl3anc 1233	. . . . 5 ⊢ (𝜑 → ((((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) mod 𝑁) = (∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) mod 𝑁) ↔ 𝑁 ∥ (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
95	4, 94	mpbid 146	. . . 4 ⊢ (𝜑 → 𝑁 ∥ (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁)))
96	24	zcnd 9335	. . . . . . . 8 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℂ)
97	96	mulid2d 7938	. . . . . . 7 ⊢ (𝜑 → (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))
98	90, 88, 97	3eqtr4a 2229	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) = (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)))
99	98	oveq2d 5869	. . . . 5 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁)) = (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))))
100	10	zcnd 9335	. . . . . 6 ⊢ (𝜑 → (𝐴↑(ϕ‘𝑁)) ∈ ℂ)
101		ax-1cn 7867	. . . . . . 7 ⊢ 1 ∈ ℂ
102		subdir 8305	. . . . . . 7 ⊢ (((𝐴↑(ϕ‘𝑁)) ∈ ℂ ∧ 1 ∈ ℂ ∧ ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℂ) → (((𝐴↑(ϕ‘𝑁)) − 1) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))))
103	101, 102	mp3an2 1320	. . . . . 6 ⊢ (((𝐴↑(ϕ‘𝑁)) ∈ ℂ ∧ ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℂ) → (((𝐴↑(ϕ‘𝑁)) − 1) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))))
104	100, 96, 103	syl2anc 409	. . . . 5 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) − 1) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))))
105	10, 11	zsubcld 9339	. . . . . . 7 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℤ)
106	105	zcnd 9335	. . . . . 6 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℂ)
107	106, 96	mulcomd 7941	. . . . 5 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) − 1) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)))
108	99, 104, 107	3eqtr2d 2209	. . . 4 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁)) = (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)))
109	95, 108	breqtrd 4015	. . 3 ⊢ (𝜑 → 𝑁 ∥ (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)))
110	1, 2, 3	eulerthlemrprm 12183	. . 3 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)
111		coprmdvds 12046	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℤ ∧ ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℤ) → ((𝑁 ∥ (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)) ∧ (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1) → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
112	65, 24, 105, 111	syl3anc 1233	. . 3 ⊢ (𝜑 → ((𝑁 ∥ (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)) ∧ (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1) → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
113	109, 110, 112	mp2and 431	. 2 ⊢ (𝜑 → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))
114		1z 9238	. . . 4 ⊢ 1 ∈ ℤ
115		moddvds 11761	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑁)) ∈ ℤ ∧ 1 ∈ ℤ) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
116	114, 115	mp3an3 1321	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑁)) ∈ ℤ) → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
117	5, 10, 116	syl2anc 409	. 2 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
118	113, 117	mpbird 166	1 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))

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 ↦ cmpt 4050 ^◡ccnv 4610 ∘ ccom 4615 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 0cc0 7774 1c1 7775 · cmul 7779 − cmin 8090 ℕcn 8878 ℕ₀cn0 9135 ℤcz 9212 ...cfz 9965 ..^cfzo 10098 mod cmo 10278 ↑cexp 10475 ∏cprod 11513 ∥ cdvds 11749 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-stab 826 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: eulerth 12187

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