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Theorem eulerthlemth 12797

Description: Lemma for eulerth 12798. 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 12795	. . . . 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 1033	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	1	simp2d 1034	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	5	phicld 12783	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
8	7	nnnn0d 9448	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
9		zexpcl 10809	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
10	6, 8, 9	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
11		1zzd 9499	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
12	7	nnzd 9594	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
13	11, 12	fzfigd 10686	. . . . . . . 8 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ∈ Fin)
14		ssrab2 3310	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
15	2, 14	eqsstri 3257	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (0..^𝑁)
16		fzo0ssnn0 10453	. . . . . . . . . . 11 ⊢ (0..^𝑁) ⊆ ℕ₀
17		nn0ssz 9490	. . . . . . . . . . 11 ⊢ ℕ₀ ⊆ ℤ
18	16, 17	sstri 3234	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℤ
19	15, 18	sstri 3234	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℤ
20		f1of 5580	. . . . . . . . . . 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	ffvelcdmda 5778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑥) ∈ 𝑆)
23	19, 22	sselid 3223	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑥) ∈ ℤ)
24	13, 23	fprodzcl 12163	. . . . . . 7 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℤ)
25	10, 24	zmulcld 9601	. . . . . 6 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) ∈ ℤ)
26		fveq2 5635	. . . . . . . . 9 ⊢ (𝑧 = (^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁)) → (𝐹‘𝑧) = (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
27		eqid 2229	. . . . . . . . . 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 12796	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
29		eqid 2229	. . . . . . . . . . . . 13 ⊢ (1...(ϕ‘𝑁)) = (1...(ϕ‘𝑁))
30		fveq2 5635	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
31	30	oveq2d 6029	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (𝐴 · (𝐹‘𝑣)) = (𝐴 · (𝐹‘𝑢)))
32	31	oveq1d 6028	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
33	32	cbvmptv 4183	. . . . . . . . . . . . 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 12792	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
35		fveq2 5635	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
36	35	oveq2d 6029	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝐴 · (𝐹‘𝑣)) = (𝐴 · (𝐹‘𝑦)))
37	36	oveq1d 6028	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) = ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
38	37	cbvmptv 4183	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
39	38	feq1i 5472	. . . . . . . . . . . 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 122	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
41		fvco3 5713	. . . . . . . . . . 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 283	. . . . . . . . . 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 2229	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
44		fveq2 5635	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
45	44	oveq2d 6029	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑥)))
46	45	oveq1d 6028	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝐴 · (𝐹‘𝑦)) mod 𝑁) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
47		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑥 ∈ (1...(ϕ‘𝑁)))
48	6	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝐴 ∈ ℤ)
49	48, 23	zmulcld 9601	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐴 · (𝐹‘𝑥)) ∈ ℤ)
50	5	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑁 ∈ ℕ)
51		zmodfzo 10602	. . . . . . . . . . . . 13 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
52	49, 50, 51	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
53	43, 46, 47, 52	fvmptd3 5736	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑥) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
54	53	fveq2d 5639	. . . . . . . . . 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 2262	. . . . . . . . 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	ffvelcdmda 5778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ 𝑆)
57	19, 56	sselid 3223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ ℤ)
58	57	zcnd 9596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ ℂ)
59	26, 13, 28, 55, 58	fprodf1o 12142	. . . . . . . 8 ⊢ (𝜑 → ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
60	3	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
61		modgcd 12555	. . . . . . . . . . . . 13 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
62	49, 50, 61	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
63	50	nnzd 9594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
64	63, 49	gcdcomd 12538	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
65	5	nnzd 9594	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
66	6, 65	gcdcomd 12538	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴))
67	1	simp3d 1035	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
68	66, 67	eqtr3d 2264	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
69	68	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd 𝐴) = 1)
70	23, 63	gcdcomd 12538	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) gcd 𝑁) = (𝑁 gcd (𝐹‘𝑥)))
71		oveq1 6020	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 gcd 𝑁) = ((𝐹‘𝑥) gcd 𝑁))
72	71	eqeq1d 2238	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘𝑥) gcd 𝑁) = 1))
73	72, 2	elrab2 2963	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ∈ 𝑆 ↔ ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
74	22, 73	sylib 122	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) ∈ (0..^𝑁) ∧ ((𝐹‘𝑥) gcd 𝑁) = 1))
75	74	simprd 114	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) gcd 𝑁) = 1)
76	70, 75	eqtr3d 2264	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐹‘𝑥)) = 1)
77		rpmul 12663	. . . . . . . . . . . . . 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 1271	. . . . . . . . . . . . 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 433	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1)
80	62, 64, 79	3eqtr2d 2268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1)
81		oveq1 6020	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → (𝑦 gcd 𝑁) = (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁))
82	81	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
83	82, 2	elrab2 2963	. . . . . . . . . . 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 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆)
85		f1ocnvfv2 5914	. . . . . . . . . 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 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
87	86	prodeq2dv 12120	. . . . . . . 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 2263	. . . . . . 7 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) = ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧))
89		fveq2 5635	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
90	89	cbvprodv 12113	. . . . . . . 8 ⊢ ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)
91	90, 24	eqeltrid 2316	. . . . . . 7 ⊢ (𝜑 → ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) ∈ ℤ)
92	88, 91	eqeltrd 2306	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ ℤ)
93		moddvds 12353	. . . . . 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 1271	. . . . 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 147	. . . 4 ⊢ (𝜑 → 𝑁 ∥ (((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) − ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁)))
96	24	zcnd 9596	. . . . . . . 8 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℂ)
97	96	mulid2d 8191	. . . . . . 7 ⊢ (𝜑 → (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))
98	90, 88, 97	3eqtr4a 2288	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) = (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)))
99	98	oveq2d 6029	. . . . 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 9596	. . . . . 6 ⊢ (𝜑 → (𝐴↑(ϕ‘𝑁)) ∈ ℂ)
101		ax-1cn 8118	. . . . . . 7 ⊢ 1 ∈ ℂ
102		subdir 8558	. . . . . . 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 1359	. . . . . 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 411	. . . . 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 9600	. . . . . . 7 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℤ)
106	105	zcnd 9596	. . . . . 6 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℂ)
107	106, 96	mulcomd 8194	. . . . 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 2268	. . . 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 4112	. . 3 ⊢ (𝜑 → 𝑁 ∥ (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)))
110	1, 2, 3	eulerthlemrprm 12794	. . 3 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)
111		coprmdvds 12657	. . . 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 1271	. . 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 433	. 2 ⊢ (𝜑 → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))
114		1z 9498	. . . 4 ⊢ 1 ∈ ℤ
115		moddvds 12353	. . . 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 1360	. . 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 411	. 2 ⊢ (𝜑 → (((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1)))
118	113, 117	mpbird 167	1 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) mod 𝑁) = (1 mod 𝑁))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 {crab 2512 class class class wbr 4086 ↦ cmpt 4148 ^◡ccnv 4722 ∘ ccom 4727 ⟶wf 5320 –1-1-onto→wf1o 5323 ‘cfv 5324 (class class class)co 6013 ℂcc 8023 0cc0 8025 1c1 8026 · cmul 8030 − cmin 8343 ℕcn 9136 ℕ₀cn0 9395 ℤcz 9472 ...cfz 10236 ..^cfzo 10370 mod cmo 10577 ↑cexp 10793 ∏cprod 12104 ∥ cdvds 12341 gcd cgcd 12517 ϕcphi 12774

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7177 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-proddc 12105 df-dvds 12342 df-gcd 12518 df-phi 12776

This theorem is referenced by: eulerth 12798

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