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Theorem eulerthlemth 13012

Description: Lemma for eulerth 13013. 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 13010	. . . . 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 1040	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	1	simp2d 1041	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	5	phicld 12998	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
8	7	nnnn0d 9622	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ₀)
9		zexpcl 10993	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑁) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
10	6, 8, 9	syl2anc 415	. . . . . . 7 ⊢ (𝜑 → (𝐴↑(ϕ‘𝑁)) ∈ ℤ)
11		1zzd 9673	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
12	7	nnzd 9769	. . . . . . . . 9 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
13	11, 12	fzfigd 10870	. . . . . . . 8 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ∈ Fin)
14		ssrab2 3333	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
15	2, 14	eqsstri 3280	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (0..^𝑁)
16		fzo0ssnn0 10635	. . . . . . . . . . 11 ⊢ (0..^𝑁) ⊆ ℕ₀
17		nn0ssz 9664	. . . . . . . . . . 11 ⊢ ℕ₀ ⊆ ℤ
18	16, 17	sstri 3257	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℤ
19	15, 18	sstri 3257	. . . . . . . . 9 ⊢ 𝑆 ⊆ ℤ
20		f1of 5639	. . . . . . . . . . 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 5843	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑥) ∈ 𝑆)
23	19, 22	sselid 3246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑥) ∈ ℤ)
24	13, 23	fprodzcl 12378	. . . . . . 7 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℤ)
25	10, 24	zmulcld 9776	. . . . . 6 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) ∈ ℤ)
26		fveq2 5695	. . . . . . . . 9 ⊢ (𝑧 = (^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁)) → (𝐹‘𝑧) = (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))))
27		eqid 2238	. . . . . . . . . 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 13011	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
29		eqid 2238	. . . . . . . . . . . . 13 ⊢ (1...(ϕ‘𝑁)) = (1...(ϕ‘𝑁))
30		fveq2 5695	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
31	30	oveq2d 6101	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → (𝐴 · (𝐹‘𝑣)) = (𝐴 · (𝐹‘𝑢)))
32	31	oveq1d 6100	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑢 → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
33	32	cbvmptv 4227	. . . . . . . . . . . . 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 13007	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
35		fveq2 5695	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
36	35	oveq2d 6101	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝐴 · (𝐹‘𝑣)) = (𝐴 · (𝐹‘𝑦)))
37	36	oveq1d 6100	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) = ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
38	37	cbvmptv 4227	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑣)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
39	38	feq1i 5526	. . . . . . . . . . . 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 5776	. . . . . . . . . . 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 2238	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
44		fveq2 5695	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
45	44	oveq2d 6101	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑥)))
46	45	oveq1d 6100	. . . . . . . . . . . 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 9776	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐴 · (𝐹‘𝑥)) ∈ ℤ)
50	5	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑁 ∈ ℕ)
51		zmodfzo 10786	. . . . . . . . . . . . 13 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
52	49, 50, 51	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ (0..^𝑁))
53	43, 46, 47, 52	fvmptd3 5799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑥) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
54	53	fveq2d 5699	. . . . . . . . . 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 2271	. . . . . . . . 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 5843	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ 𝑆)
57	19, 56	sselid 3246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ ℤ)
58	57	zcnd 9771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑧) ∈ ℂ)
59	26, 13, 28, 55, 58	fprodf1o 12357	. . . . . . . 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 12770	. . . . . . . . . . . . 13 ⊢ (((𝐴 · (𝐹‘𝑥)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
62	49, 50, 61	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
63	50	nnzd 9769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → 𝑁 ∈ ℤ)
64	63, 49	gcdcomd 12753	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = ((𝐴 · (𝐹‘𝑥)) gcd 𝑁))
65	5	nnzd 9769	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℤ)
66	6, 65	gcdcomd 12753	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 gcd 𝑁) = (𝑁 gcd 𝐴))
67	1	simp3d 1042	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
68	66, 67	eqtr3d 2273	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
69	68	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd 𝐴) = 1)
70	23, 63	gcdcomd 12753	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐹‘𝑥) gcd 𝑁) = (𝑁 gcd (𝐹‘𝑥)))
71		oveq1 6092	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 gcd 𝑁) = ((𝐹‘𝑥) gcd 𝑁))
72	71	eqeq1d 2247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦 gcd 𝑁) = 1 ↔ ((𝐹‘𝑥) gcd 𝑁) = 1))
73	72, 2	elrab2 2985	. . . . . . . . . . . . . . . 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 2273	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐹‘𝑥)) = 1)
77		rpmul 12878	. . . . . . . . . . . . . 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 1278	. . . . . . . . . . . . 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 437	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝑁 gcd (𝐴 · (𝐹‘𝑥))) = 1)
80	62, 64, 79	3eqtr2d 2277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1)
81		oveq1 6092	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → (𝑦 gcd 𝑁) = (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁))
82	81	eqeq1d 2247	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝐴 · (𝐹‘𝑥)) mod 𝑁) → ((𝑦 gcd 𝑁) = 1 ↔ (((𝐴 · (𝐹‘𝑥)) mod 𝑁) gcd 𝑁) = 1))
83	82, 2	elrab2 2985	. . . . . . . . . . 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 421	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → ((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ 𝑆)
85		f1ocnvfv2 5984	. . . . . . . . . 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 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(ϕ‘𝑁))) → (𝐹‘(^◡𝐹‘((𝐴 · (𝐹‘𝑥)) mod 𝑁))) = ((𝐴 · (𝐹‘𝑥)) mod 𝑁))
87	86	prodeq2dv 12335	. . . . . . . 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 2272	. . . . . . 7 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) = ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧))
89		fveq2 5695	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
90	89	cbvprodv 12328	. . . . . . . 8 ⊢ ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)
91	90, 24	eqeltrid 2325	. . . . . . 7 ⊢ (𝜑 → ∏𝑧 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑧) ∈ ℤ)
92	88, 91	eqeltrd 2315	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) ∈ ℤ)
93		moddvds 12568	. . . . . 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 1278	. . . . 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 9771	. . . . . . . 8 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) ∈ ℂ)
97	96	mullidd 8344	. . . . . . 7 ⊢ (𝜑 → (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥))
98	90, 88, 97	3eqtr4a 2297	. . . . . 6 ⊢ (𝜑 → ∏𝑥 ∈ (1...(ϕ‘𝑁))((𝐴 · (𝐹‘𝑥)) mod 𝑁) = (1 · ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)))
99	98	oveq2d 6101	. . . . 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 9771	. . . . . 6 ⊢ (𝜑 → (𝐴↑(ϕ‘𝑁)) ∈ ℂ)
101		ax-1cn 8272	. . . . . . 7 ⊢ 1 ∈ ℂ
102		subdir 8713	. . . . . . 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 1366	. . . . . 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 415	. . . . 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 9775	. . . . . . 7 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℤ)
106	105	zcnd 9771	. . . . . 6 ⊢ (𝜑 → ((𝐴↑(ϕ‘𝑁)) − 1) ∈ ℂ)
107	106, 96	mulcomd 8347	. . . . 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 2277	. . . 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 4156	. . 3 ⊢ (𝜑 → 𝑁 ∥ (∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥) · ((𝐴↑(ϕ‘𝑁)) − 1)))
110	1, 2, 3	eulerthlemrprm 13009	. . 3 ⊢ (𝜑 → (𝑁 gcd ∏𝑥 ∈ (1...(ϕ‘𝑁))(𝐹‘𝑥)) = 1)
111		coprmdvds 12872	. . . 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 1278	. . 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 437	. 2 ⊢ (𝜑 → 𝑁 ∥ ((𝐴↑(ϕ‘𝑁)) − 1))
114		1z 9672	. . . 4 ⊢ 1 ∈ ℤ
115		moddvds 12568	. . . 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 1367	. . 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 415	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 {crab 2532 class class class wbr 4130 ↦ cmpt 4192 ^◡ccnv 4773 ∘ ccom 4778 ⟶wf 5373 –1-1-onto→wf1o 5376 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 0cc0 8179 1c1 8180 · cmul 8184 − cmin 8497 ℕcn 9305 ℕ₀cn0 9565 ℤcz 9646 ...cfz 10413 ..^cfzo 10551 mod cmo 10761 ↑cexp 10977 ∏cprod 12319 ∥ cdvds 12556 gcd cgcd 12732 ϕcphi 12989

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-fz 10414 df-fzo 10552 df-fl 10707 df-mod 10762 df-seqfrec 10887 df-exp 10978 df-ihash 11217 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767 df-clim 12047 df-proddc 12320 df-dvds 12557 df-gcd 12733 df-phi 12991

This theorem is used by: eulerth 13013

W3C validator