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Theorem eulerthlemh 12231

Description: Lemma for eulerth 12233. A permutation of (1...(ϕ‘𝑁)). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 5-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)
eulerth.h	⊢ 𝐻 = (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))

**Assertion**
Ref	Expression
eulerthlemh	⊢ (𝜑 → 𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))

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

Proof of Theorem eulerthlemh Dummy variables 𝑎 𝑏 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eulerth.4	. . . 4 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆)
2		f1ocnv 5475	. . . 4 ⊢ (𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆 → ^◡𝐹:𝑆–1-1-onto→(1...(ϕ‘𝑁)))
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → ^◡𝐹:𝑆–1-1-onto→(1...(ϕ‘𝑁)))
4		eulerth.1	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1))
5		eulerth.2	. . . . . . 7 ⊢ 𝑆 = {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1}
6		eqid 2177	. . . . . . 7 ⊢ (1...(ϕ‘𝑁)) = (1...(ϕ‘𝑁))
7		fveq2 5516	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
8	7	oveq2d 5891	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝐴 · (𝐹‘𝑎)) = (𝐴 · (𝐹‘𝑏)))
9	8	oveq1d 5890	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝐴 · (𝐹‘𝑎)) mod 𝑁) = ((𝐴 · (𝐹‘𝑏)) mod 𝑁))
10	9	cbvmptv 4100	. . . . . . 7 ⊢ (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)) = (𝑏 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑏)) mod 𝑁))
11	4, 5, 6, 1, 10	eulerthlem1 12227	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
12		fveq2 5516	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
13	12	oveq2d 5891	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (𝐴 · (𝐹‘𝑎)) = (𝐴 · (𝐹‘𝑦)))
14	13	oveq1d 5890	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → ((𝐴 · (𝐹‘𝑎)) mod 𝑁) = ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
15	14	cbvmptv 4100	. . . . . . 7 ⊢ (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
16	15	feq1i 5359	. . . . . 6 ⊢ ((𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆 ↔ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
17	11, 16	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
18	4	simp1d 1009	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑁 ∈ ℕ)
20	4	simp2d 1010	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
21	20	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐴 ∈ ℤ)
22		ssrab2 3241	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
23	5, 22	eqsstri 3188	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ (0..^𝑁)
24		fzo0ssnn0 10215	. . . . . . . . . . . . 13 ⊢ (0..^𝑁) ⊆ ℕ₀
25		nn0ssz 9271	. . . . . . . . . . . . 13 ⊢ ℕ₀ ⊆ ℤ
26	24, 25	sstri 3165	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ⊆ ℤ
27	23, 26	sstri 3165	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ℤ
28		f1of 5462	. . . . . . . . . . . . . 14 ⊢ (𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆 → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
29	1, 28	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
30	29	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐹:(1...(ϕ‘𝑁))⟶𝑆)
31		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑢 ∈ (1...(ϕ‘𝑁)))
32	30, 31	ffvelcdmd 5653	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ 𝑆)
33	27, 32	sselid 3154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℤ)
34	21, 33	zmulcld 9381	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · (𝐹‘𝑢)) ∈ ℤ)
35	29	ffvelcdmda 5652	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑣 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑣) ∈ 𝑆)
36	35	adantrl 478	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ 𝑆)
37	27, 36	sselid 3154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℤ)
38	21, 37	zmulcld 9381	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · (𝐹‘𝑣)) ∈ ℤ)
39		moddvds 11806	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 · (𝐹‘𝑢)) ∈ ℤ ∧ (𝐴 · (𝐹‘𝑣)) ∈ ℤ) → (((𝐴 · (𝐹‘𝑢)) mod 𝑁) = ((𝐴 · (𝐹‘𝑣)) mod 𝑁) ↔ 𝑁 ∥ ((𝐴 · (𝐹‘𝑢)) − (𝐴 · (𝐹‘𝑣)))))
40	19, 34, 38, 39	syl3anc 1238	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝐴 · (𝐹‘𝑢)) mod 𝑁) = ((𝐴 · (𝐹‘𝑣)) mod 𝑁) ↔ 𝑁 ∥ ((𝐴 · (𝐹‘𝑢)) − (𝐴 · (𝐹‘𝑣)))))
41		eqid 2177	. . . . . . . . . 10 ⊢ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
42		fveq2 5516	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
43	42	oveq2d 5891	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑢)))
44	43	oveq1d 5890	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → ((𝐴 · (𝐹‘𝑦)) mod 𝑁) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
45		zmodfzo 10347	. . . . . . . . . . 11 ⊢ (((𝐴 · (𝐹‘𝑢)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑢)) mod 𝑁) ∈ (0..^𝑁))
46	34, 19, 45	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐴 · (𝐹‘𝑢)) mod 𝑁) ∈ (0..^𝑁))
47	41, 44, 31, 46	fvmptd3 5610	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑢) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
48		fveq2 5516	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
49	48	oveq2d 5891	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑣)))
50	49	oveq1d 5890	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ((𝐴 · (𝐹‘𝑦)) mod 𝑁) = ((𝐴 · (𝐹‘𝑣)) mod 𝑁))
51		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑣 ∈ (1...(ϕ‘𝑁)))
52		zmodfzo 10347	. . . . . . . . . . 11 ⊢ (((𝐴 · (𝐹‘𝑣)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) ∈ (0..^𝑁))
53	38, 19, 52	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐴 · (𝐹‘𝑣)) mod 𝑁) ∈ (0..^𝑁))
54	41, 50, 51, 53	fvmptd3 5610	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑣) = ((𝐴 · (𝐹‘𝑣)) mod 𝑁))
55	47, 54	eqeq12d 2192	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑢) = ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑣) ↔ ((𝐴 · (𝐹‘𝑢)) mod 𝑁) = ((𝐴 · (𝐹‘𝑣)) mod 𝑁)))
56	21	zcnd 9376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐴 ∈ ℂ)
57	33	zcnd 9376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℂ)
58	37	zcnd 9376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℂ)
59	56, 57, 58	subdid 8371	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) = ((𝐴 · (𝐹‘𝑢)) − (𝐴 · (𝐹‘𝑣))))
60	59	breq2d 4016	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝑁 ∥ (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) ↔ 𝑁 ∥ ((𝐴 · (𝐹‘𝑢)) − (𝐴 · (𝐹‘𝑣)))))
61	40, 55, 60	3bitr4d 220	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑢) = ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑣) ↔ 𝑁 ∥ (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣)))))
62	18	nnzd 9374	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
63	62, 20	gcdcomd 11975	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝐴) = (𝐴 gcd 𝑁))
64	4	simp3d 1011	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
65	63, 64	eqtrd 2210	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 gcd 𝐴) = 1)
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝑁 gcd 𝐴) = 1)
67	62	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑁 ∈ ℤ)
68	33, 37	zsubcld 9380	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) − (𝐹‘𝑣)) ∈ ℤ)
69		coprmdvds 12092	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ ((𝐹‘𝑢) − (𝐹‘𝑣)) ∈ ℤ) → ((𝑁 ∥ (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) ∧ (𝑁 gcd 𝐴) = 1) → 𝑁 ∥ ((𝐹‘𝑢) − (𝐹‘𝑣))))
70	67, 21, 68, 69	syl3anc 1238	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝑁 ∥ (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) ∧ (𝑁 gcd 𝐴) = 1) → 𝑁 ∥ ((𝐹‘𝑢) − (𝐹‘𝑣))))
71		zq 9626	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑢) ∈ ℤ → (𝐹‘𝑢) ∈ ℚ)
72	33, 71	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℚ)
73		zq 9626	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
74	62, 73	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℚ)
75	74	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑁 ∈ ℚ)
76	23, 32	sselid 3154	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ (0..^𝑁))
77		elfzole1 10155	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑢) ∈ (0..^𝑁) → 0 ≤ (𝐹‘𝑢))
78	76, 77	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 0 ≤ (𝐹‘𝑢))
79		elfzolt2 10156	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑢) ∈ (0..^𝑁) → (𝐹‘𝑢) < 𝑁)
80	76, 79	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) < 𝑁)
81		modqid 10349	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑢) ∈ ℚ ∧ 𝑁 ∈ ℚ) ∧ (0 ≤ (𝐹‘𝑢) ∧ (𝐹‘𝑢) < 𝑁)) → ((𝐹‘𝑢) mod 𝑁) = (𝐹‘𝑢))
82	72, 75, 78, 80, 81	syl22anc 1239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) mod 𝑁) = (𝐹‘𝑢))
83	27, 35	sselid 3154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑣) ∈ ℤ)
84	83	adantrl 478	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℤ)
85		zq 9626	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑣) ∈ ℤ → (𝐹‘𝑣) ∈ ℚ)
86	84, 85	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℚ)
87	23, 35	sselid 3154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑣) ∈ (0..^𝑁))
88		elfzole1 10155	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑣) ∈ (0..^𝑁) → 0 ≤ (𝐹‘𝑣))
89	87, 88	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑣 ∈ (1...(ϕ‘𝑁))) → 0 ≤ (𝐹‘𝑣))
90	89	adantrl 478	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 0 ≤ (𝐹‘𝑣))
91	87	adantrl 478	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ (0..^𝑁))
92		elfzolt2 10156	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑣) ∈ (0..^𝑁) → (𝐹‘𝑣) < 𝑁)
93	91, 92	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) < 𝑁)
94		modqid 10349	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑣) ∈ ℚ ∧ 𝑁 ∈ ℚ) ∧ (0 ≤ (𝐹‘𝑣) ∧ (𝐹‘𝑣) < 𝑁)) → ((𝐹‘𝑣) mod 𝑁) = (𝐹‘𝑣))
95	86, 75, 90, 93, 94	syl22anc 1239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑣) mod 𝑁) = (𝐹‘𝑣))
96	82, 95	eqeq12d 2192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝐹‘𝑢) mod 𝑁) = ((𝐹‘𝑣) mod 𝑁) ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
97		moddvds 11806	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐹‘𝑢) ∈ ℤ ∧ (𝐹‘𝑣) ∈ ℤ) → (((𝐹‘𝑢) mod 𝑁) = ((𝐹‘𝑣) mod 𝑁) ↔ 𝑁 ∥ ((𝐹‘𝑢) − (𝐹‘𝑣))))
98	19, 33, 37, 97	syl3anc 1238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝐹‘𝑢) mod 𝑁) = ((𝐹‘𝑣) mod 𝑁) ↔ 𝑁 ∥ ((𝐹‘𝑢) − (𝐹‘𝑣))))
99		f1of1 5461	. . . . . . . . . . . 12 ⊢ (𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆 → 𝐹:(1...(ϕ‘𝑁))–1-1→𝑆)
100	1, 99	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:(1...(ϕ‘𝑁))–1-1→𝑆)
101		f1fveq 5773	. . . . . . . . . . 11 ⊢ ((𝐹:(1...(ϕ‘𝑁))–1-1→𝑆 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
102	100, 101	sylan 283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
103	96, 98, 102	3bitr3d 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝑁 ∥ ((𝐹‘𝑢) − (𝐹‘𝑣)) ↔ 𝑢 = 𝑣))
104	70, 103	sylibd 149	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝑁 ∥ (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) ∧ (𝑁 gcd 𝐴) = 1) → 𝑢 = 𝑣))
105	66, 104	mpan2d 428	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝑁 ∥ (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) → 𝑢 = 𝑣))
106	61, 105	sylbid 150	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑢) = ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑣) → 𝑢 = 𝑣))
107	106	ralrimivva 2559	. . . . 5 ⊢ (𝜑 → ∀𝑢 ∈ (1...(ϕ‘𝑁))∀𝑣 ∈ (1...(ϕ‘𝑁))(((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑢) = ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑣) → 𝑢 = 𝑣))
108		dff13 5769	. . . . 5 ⊢ ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1→𝑆 ↔ ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆 ∧ ∀𝑢 ∈ (1...(ϕ‘𝑁))∀𝑣 ∈ (1...(ϕ‘𝑁))(((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑢) = ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))‘𝑣) → 𝑢 = 𝑣)))
109	17, 107, 108	sylanbrc 417	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1→𝑆)
110		1zzd 9280	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
111	18	phicld 12218	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
112	111	nnzd 9374	. . . . . . 7 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
113	110, 112	fzfigd 10431	. . . . . 6 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ∈ Fin)
114		f1oeng 6757	. . . . . 6 ⊢ (((1...(ϕ‘𝑁)) ∈ Fin ∧ 𝐹:(1...(ϕ‘𝑁))–1-1-onto→𝑆) → (1...(ϕ‘𝑁)) ≈ 𝑆)
115	113, 1, 114	syl2anc 411	. . . . 5 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ≈ 𝑆)
116	4, 5	eulerthlemfi 12228	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Fin)
117		f1finf1o 6946	. . . . 5 ⊢ (((1...(ϕ‘𝑁)) ≈ 𝑆 ∧ 𝑆 ∈ Fin) → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1→𝑆 ↔ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1-onto→𝑆))
118	115, 116, 117	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1→𝑆 ↔ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1-onto→𝑆))
119	109, 118	mpbid 147	. . 3 ⊢ (𝜑 → (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1-onto→𝑆)
120		f1oco 5485	. . 3 ⊢ ((^◡𝐹:𝑆–1-1-onto→(1...(ϕ‘𝑁)) ∧ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)):(1...(ϕ‘𝑁))–1-1-onto→𝑆) → (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
121	3, 119, 120	syl2anc 411	. 2 ⊢ (𝜑 → (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
122		eulerth.h	. . 3 ⊢ 𝐻 = (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)))
123		f1oeq1 5450	. . 3 ⊢ (𝐻 = (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))) → (𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)) ↔ (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁))))
124	122, 123	ax-mp 5	. 2 ⊢ (𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)) ↔ (^◡𝐹 ∘ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))):(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))
125	121, 124	sylibr 134	1 ⊢ (𝜑 → 𝐻:(1...(ϕ‘𝑁))–1-1-onto→(1...(ϕ‘𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {crab 2459 class class class wbr 4004 ↦ cmpt 4065 ^◡ccnv 4626 ∘ ccom 4631 ⟶wf 5213 –1-1→wf1 5214 –1-1-onto→wf1o 5216 ‘cfv 5217 (class class class)co 5875 ≈ cen 6738 Fincfn 6740 0cc0 7811 1c1 7812 · cmul 7816 < clt 7992 ≤ cle 7993 − cmin 8128 ℕcn 8919 ℕ₀cn0 9176 ℤcz 9253 ℚcq 9619 ...cfz 10008 ..^cfzo 10142 mod cmo 10322 ∥ cdvds 11794 gcd cgcd 11943 ϕcphi 12209

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-1o 6417 df-er 6535 df-en 6741 df-dom 6742 df-fin 6743 df-sup 6983 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-fl 10270 df-mod 10323 df-seqfrec 10446 df-exp 10520 df-ihash 10756 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-dvds 11795 df-gcd 11944 df-phi 12211

This theorem is referenced by: eulerthlemth 12232

W3C validator