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Theorem eulerthlemh 12719

Description: Lemma for eulerth 12721. 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 5561	. . . 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 2209	. . . . . . 7 ⊢ (1...(ϕ‘𝑁)) = (1...(ϕ‘𝑁))
7		fveq2 5603	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
8	7	oveq2d 5990	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝐴 · (𝐹‘𝑎)) = (𝐴 · (𝐹‘𝑏)))
9	8	oveq1d 5989	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝐴 · (𝐹‘𝑎)) mod 𝑁) = ((𝐴 · (𝐹‘𝑏)) mod 𝑁))
10	9	cbvmptv 4159	. . . . . . 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 12715	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
12		fveq2 5603	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
13	12	oveq2d 5990	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (𝐴 · (𝐹‘𝑎)) = (𝐴 · (𝐹‘𝑦)))
14	13	oveq1d 5989	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → ((𝐴 · (𝐹‘𝑎)) mod 𝑁) = ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
15	14	cbvmptv 4159	. . . . . . 7 ⊢ (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
16	15	feq1i 5442	. . . . . 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 1014	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑁 ∈ ℕ)
20	4	simp2d 1015	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
21	20	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐴 ∈ ℤ)
22		ssrab2 3289	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
23	5, 22	eqsstri 3236	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ (0..^𝑁)
24		fzo0ssnn0 10388	. . . . . . . . . . . . 13 ⊢ (0..^𝑁) ⊆ ℕ₀
25		nn0ssz 9432	. . . . . . . . . . . . 13 ⊢ ℕ₀ ⊆ ℤ
26	24, 25	sstri 3213	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ⊆ ℤ
27	23, 26	sstri 3213	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ℤ
28		f1of 5548	. . . . . . . . . . . . . 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 5744	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ 𝑆)
33	27, 32	sselid 3202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℤ)
34	21, 33	zmulcld 9543	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · (𝐹‘𝑢)) ∈ ℤ)
35	29	ffvelcdmda 5743	. . . . . . . . . . . 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 3202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℤ)
38	21, 37	zmulcld 9543	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · (𝐹‘𝑣)) ∈ ℤ)
39		moddvds 12276	. . . . . . . . 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 1252	. . . . . . . 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 2209	. . . . . . . . . 10 ⊢ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
42		fveq2 5603	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
43	42	oveq2d 5990	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑢)))
44	43	oveq1d 5989	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → ((𝐴 · (𝐹‘𝑦)) mod 𝑁) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
45		zmodfzo 10536	. . . . . . . . . . 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 5701	. . . . . . . . 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 5603	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
49	48	oveq2d 5990	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑣)))
50	49	oveq1d 5989	. . . . . . . . . 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 10536	. . . . . . . . . . 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 5701	. . . . . . . . 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 2224	. . . . . . . 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 9538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐴 ∈ ℂ)
57	33	zcnd 9538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℂ)
58	37	zcnd 9538	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℂ)
59	56, 57, 58	subdid 8528	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) = ((𝐴 · (𝐹‘𝑢)) − (𝐴 · (𝐹‘𝑣))))
60	59	breq2d 4074	. . . . . . . 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 9536	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
63	62, 20	gcdcomd 12461	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝐴) = (𝐴 gcd 𝑁))
64	4	simp3d 1016	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
65	63, 64	eqtrd 2242	. . . . . . . . 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 9542	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) − (𝐹‘𝑣)) ∈ ℤ)
69		coprmdvds 12580	. . . . . . . . . 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 1252	. . . . . . . . 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 9789	. . . . . . . . . . . . 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 9789	. . . . . . . . . . . . . 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 3202	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ (0..^𝑁))
77		elfzole1 10320	. . . . . . . . . . . . 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 10321	. . . . . . . . . . . . 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 10538	. . . . . . . . . . . 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 1253	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) mod 𝑁) = (𝐹‘𝑢))
83	27, 35	sselid 3202	. . . . . . . . . . . . . 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 9789	. . . . . . . . . . . . 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 3202	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑣) ∈ (0..^𝑁))
88		elfzole1 10320	. . . . . . . . . . . . . 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 10321	. . . . . . . . . . . . 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 10538	. . . . . . . . . . . 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 1253	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑣) mod 𝑁) = (𝐹‘𝑣))
96	82, 95	eqeq12d 2224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝐹‘𝑢) mod 𝑁) = ((𝐹‘𝑣) mod 𝑁) ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
97		moddvds 12276	. . . . . . . . . . 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 1252	. . . . . . . . . 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 5547	. . . . . . . . . . . 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 5869	. . . . . . . . . . 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 2592	. . . . 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 5865	. . . . 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 9441	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
111	18	phicld 12706	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
112	111	nnzd 9536	. . . . . . 7 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
113	110, 112	fzfigd 10620	. . . . . 6 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ∈ Fin)
114		f1oeng 6878	. . . . . 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 12716	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Fin)
117		f1finf1o 7082	. . . . 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 5571	. . 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 5536	. . 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 983 = wceq 1375 ∈ wcel 2180 ∀wral 2488 {crab 2492 class class class wbr 4062 ↦ cmpt 4124 ^◡ccnv 4695 ∘ ccom 4700 ⟶wf 5290 –1-1→wf1 5291 –1-1-onto→wf1o 5293 ‘cfv 5294 (class class class)co 5974 ≈ cen 6855 Fincfn 6857 0cc0 7967 1c1 7968 · cmul 7972 < clt 8149 ≤ cle 8150 − cmin 8285 ℕcn 9078 ℕ₀cn0 9337 ℤcz 9414 ℚcq 9782 ...cfz 10172 ..^cfzo 10306 mod cmo 10511 ∥ cdvds 12264 gcd cgcd 12440 ϕcphi 12697

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-stab 835 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-frec 6507 df-1o 6532 df-er 6650 df-en 6858 df-dom 6859 df-fin 6860 df-sup 7119 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-fz 10173 df-fzo 10307 df-fl 10457 df-mod 10512 df-seqfrec 10637 df-exp 10728 df-ihash 10965 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-dvds 12265 df-gcd 12441 df-phi 12699

This theorem is referenced by: eulerthlemth 12720

W3C validator