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Theorem eulerthlemh 12399

Description: Lemma for eulerth 12401. 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 5517	. . . 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 2196	. . . . . . 7 ⊢ (1...(ϕ‘𝑁)) = (1...(ϕ‘𝑁))
7		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
8	7	oveq2d 5938	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝐴 · (𝐹‘𝑎)) = (𝐴 · (𝐹‘𝑏)))
9	8	oveq1d 5937	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝐴 · (𝐹‘𝑎)) mod 𝑁) = ((𝐴 · (𝐹‘𝑏)) mod 𝑁))
10	9	cbvmptv 4129	. . . . . . 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 12395	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)):(1...(ϕ‘𝑁))⟶𝑆)
12		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑎 = 𝑦 → (𝐹‘𝑎) = (𝐹‘𝑦))
13	12	oveq2d 5938	. . . . . . . . 9 ⊢ (𝑎 = 𝑦 → (𝐴 · (𝐹‘𝑎)) = (𝐴 · (𝐹‘𝑦)))
14	13	oveq1d 5937	. . . . . . . 8 ⊢ (𝑎 = 𝑦 → ((𝐴 · (𝐹‘𝑎)) mod 𝑁) = ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
15	14	cbvmptv 4129	. . . . . . 7 ⊢ (𝑎 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑎)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
16	15	feq1i 5400	. . . . . 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 1011	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
19	18	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝑁 ∈ ℕ)
20	4	simp2d 1012	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
21	20	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐴 ∈ ℤ)
22		ssrab2 3268	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (0..^𝑁) ∣ (𝑦 gcd 𝑁) = 1} ⊆ (0..^𝑁)
23	5, 22	eqsstri 3215	. . . . . . . . . . . 12 ⊢ 𝑆 ⊆ (0..^𝑁)
24		fzo0ssnn0 10291	. . . . . . . . . . . . 13 ⊢ (0..^𝑁) ⊆ ℕ₀
25		nn0ssz 9344	. . . . . . . . . . . . 13 ⊢ ℕ₀ ⊆ ℤ
26	24, 25	sstri 3192	. . . . . . . . . . . 12 ⊢ (0..^𝑁) ⊆ ℤ
27	23, 26	sstri 3192	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ℤ
28		f1of 5504	. . . . . . . . . . . . . 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 5698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ 𝑆)
33	27, 32	sselid 3181	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℤ)
34	21, 33	zmulcld 9454	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · (𝐹‘𝑢)) ∈ ℤ)
35	29	ffvelcdmda 5697	. . . . . . . . . . . 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 3181	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℤ)
38	21, 37	zmulcld 9454	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · (𝐹‘𝑣)) ∈ ℤ)
39		moddvds 11964	. . . . . . . . 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 1249	. . . . . . . 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 2196	. . . . . . . . . 10 ⊢ (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁)) = (𝑦 ∈ (1...(ϕ‘𝑁)) ↦ ((𝐴 · (𝐹‘𝑦)) mod 𝑁))
42		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
43	42	oveq2d 5938	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑢 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑢)))
44	43	oveq1d 5937	. . . . . . . . . 10 ⊢ (𝑦 = 𝑢 → ((𝐴 · (𝐹‘𝑦)) mod 𝑁) = ((𝐴 · (𝐹‘𝑢)) mod 𝑁))
45		zmodfzo 10439	. . . . . . . . . . 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 5655	. . . . . . . . 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 5558	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
49	48	oveq2d 5938	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐴 · (𝐹‘𝑦)) = (𝐴 · (𝐹‘𝑣)))
50	49	oveq1d 5937	. . . . . . . . . 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 10439	. . . . . . . . . . 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 5655	. . . . . . . . 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 2211	. . . . . . . 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 9449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → 𝐴 ∈ ℂ)
57	33	zcnd 9449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ ℂ)
58	37	zcnd 9449	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑣) ∈ ℂ)
59	56, 57, 58	subdid 8440	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐴 · ((𝐹‘𝑢) − (𝐹‘𝑣))) = ((𝐴 · (𝐹‘𝑢)) − (𝐴 · (𝐹‘𝑣))))
60	59	breq2d 4045	. . . . . . . 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 9447	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
63	62, 20	gcdcomd 12141	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 𝐴) = (𝐴 gcd 𝑁))
64	4	simp3d 1013	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝑁) = 1)
65	63, 64	eqtrd 2229	. . . . . . . . 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 9453	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) − (𝐹‘𝑣)) ∈ ℤ)
69		coprmdvds 12260	. . . . . . . . . 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 1249	. . . . . . . . 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 9700	. . . . . . . . . . . . 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 9700	. . . . . . . . . . . . . 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 3181	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (𝐹‘𝑢) ∈ (0..^𝑁))
77		elfzole1 10231	. . . . . . . . . . . . 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 10232	. . . . . . . . . . . . 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 10441	. . . . . . . . . . . 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 1250	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑢) mod 𝑁) = (𝐹‘𝑢))
83	27, 35	sselid 3181	. . . . . . . . . . . . . 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 9700	. . . . . . . . . . . . 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 3181	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑣 ∈ (1...(ϕ‘𝑁))) → (𝐹‘𝑣) ∈ (0..^𝑁))
88		elfzole1 10231	. . . . . . . . . . . . . 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 10232	. . . . . . . . . . . . 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 10441	. . . . . . . . . . . 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 1250	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → ((𝐹‘𝑣) mod 𝑁) = (𝐹‘𝑣))
96	82, 95	eqeq12d 2211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (1...(ϕ‘𝑁)) ∧ 𝑣 ∈ (1...(ϕ‘𝑁)))) → (((𝐹‘𝑢) mod 𝑁) = ((𝐹‘𝑣) mod 𝑁) ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
97		moddvds 11964	. . . . . . . . . . 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 1249	. . . . . . . . . 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 5503	. . . . . . . . . . . 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 5819	. . . . . . . . . . 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 2579	. . . . 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 5815	. . . . 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 9353	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
111	18	phicld 12386	. . . . . . . 8 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℕ)
112	111	nnzd 9447	. . . . . . 7 ⊢ (𝜑 → (ϕ‘𝑁) ∈ ℤ)
113	110, 112	fzfigd 10523	. . . . . 6 ⊢ (𝜑 → (1...(ϕ‘𝑁)) ∈ Fin)
114		f1oeng 6816	. . . . . 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 12396	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Fin)
117		f1finf1o 7013	. . . . 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 5527	. . 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 5492	. . 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 980 = wceq 1364 ∈ wcel 2167 ∀wral 2475 {crab 2479 class class class wbr 4033 ↦ cmpt 4094 ^◡ccnv 4662 ∘ ccom 4667 ⟶wf 5254 –1-1→wf1 5255 –1-1-onto→wf1o 5257 ‘cfv 5258 (class class class)co 5922 ≈ cen 6797 Fincfn 6799 0cc0 7879 1c1 7880 · cmul 7884 < clt 8061 ≤ cle 8062 − cmin 8197 ℕcn 8990 ℕ₀cn0 9249 ℤcz 9326 ℚcq 9693 ...cfz 10083 ..^cfzo 10217 mod cmo 10414 ∥ cdvds 11952 gcd cgcd 12120 ϕcphi 12377

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-1o 6474 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-sup 7050 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121 df-phi 12379

This theorem is referenced by: eulerthlemth 12400

W3C validator