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Theorem wilthlem3 27131

Description: Lemma for wilth 27132. Here we round out the argument of wilthlem2 27130 with the final step of the induction. The induction argument shows that every subset of 1...(𝑃 − 1) that is closed under inverse and contains 𝑃 − 1 multiplies to -1 mod 𝑃, and clearly 1...(𝑃 − 1) itself is such a set. Thus, the product of all the elements is -1, and all that is left is to translate the group sum notation (which we used for its unordered summing capabilities) into an ordered sequence to match the definition of the factorial. (Contributed by Mario Carneiro, 24-Jan-2015.) (Proof shortened by AV, 27-Jul-2019.)

**Hypotheses**
Ref	Expression
wilthlem.t	⊢ 𝑇 = (mulGrp‘ℂ_fld)
wilthlem.a	⊢ 𝐴 = {𝑥 ∈ 𝒫 (1...(𝑃 − 1)) ∣ ((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}

**Assertion**
Ref	Expression
wilthlem3	⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝑃,𝑦 𝑥,𝑇,𝑦

Proof of Theorem wilthlem3 Dummy variables 𝑡 𝑠 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmuz2 16743	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
2		uz2m1nn 12988	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 − 1) ∈ ℕ)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ ℕ)
4		nnuz 12946	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
5	3, 4	eleqtrdi 2854	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ (ℤ_≥‘1))
6		eluzfz2 13592	. . . . . 6 ⊢ ((𝑃 − 1) ∈ (ℤ_≥‘1) → (𝑃 − 1) ∈ (1...(𝑃 − 1)))
7	5, 6	syl 17	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ (1...(𝑃 − 1)))
8		simpl 482	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
9		elfzelz 13584	. . . . . . . . 9 ⊢ (𝑦 ∈ (1...(𝑃 − 1)) → 𝑦 ∈ ℤ)
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → 𝑦 ∈ ℤ)
11		prmnn 16721	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
12		fzm1ndvds 16370	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑦)
13	11, 12	sylan 579	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑦)
14		eqid 2740	. . . . . . . . 9 ⊢ ((𝑦↑(𝑃 − 2)) mod 𝑃) = ((𝑦↑(𝑃 − 2)) mod 𝑃)
15	14	prmdiv 16832	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ ¬ 𝑃 ∥ 𝑦) → (((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑦 · ((𝑦↑(𝑃 − 2)) mod 𝑃)) − 1)))
16	8, 10, 13, 15	syl3anc 1371	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → (((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)) ∧ 𝑃 ∥ ((𝑦 · ((𝑦↑(𝑃 − 2)) mod 𝑃)) − 1)))
17	16	simpld 494	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)))
18	17	ralrimiva 3152	. . . . 5 ⊢ (𝑃 ∈ ℙ → ∀𝑦 ∈ (1...(𝑃 − 1))((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)))
19		ovex 7481	. . . . . . 7 ⊢ (1...(𝑃 − 1)) ∈ V
20	19	pwid 4644	. . . . . 6 ⊢ (1...(𝑃 − 1)) ∈ 𝒫 (1...(𝑃 − 1))
21		eleq2 2833	. . . . . . . 8 ⊢ (𝑥 = (1...(𝑃 − 1)) → ((𝑃 − 1) ∈ 𝑥 ↔ (𝑃 − 1) ∈ (1...(𝑃 − 1))))
22		eleq2 2833	. . . . . . . . 9 ⊢ (𝑥 = (1...(𝑃 − 1)) → (((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥 ↔ ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1))))
23	22	raleqbi1dv 3346	. . . . . . . 8 ⊢ (𝑥 = (1...(𝑃 − 1)) → (∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥 ↔ ∀𝑦 ∈ (1...(𝑃 − 1))((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1))))
24	21, 23	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = (1...(𝑃 − 1)) → (((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥) ↔ ((𝑃 − 1) ∈ (1...(𝑃 − 1)) ∧ ∀𝑦 ∈ (1...(𝑃 − 1))((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)))))
25		wilthlem.a	. . . . . . 7 ⊢ 𝐴 = {𝑥 ∈ 𝒫 (1...(𝑃 − 1)) ∣ ((𝑃 − 1) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥)}
26	24, 25	elrab2 3711	. . . . . 6 ⊢ ((1...(𝑃 − 1)) ∈ 𝐴 ↔ ((1...(𝑃 − 1)) ∈ 𝒫 (1...(𝑃 − 1)) ∧ ((𝑃 − 1) ∈ (1...(𝑃 − 1)) ∧ ∀𝑦 ∈ (1...(𝑃 − 1))((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)))))
27	20, 26	mpbiran 708	. . . . 5 ⊢ ((1...(𝑃 − 1)) ∈ 𝐴 ↔ ((𝑃 − 1) ∈ (1...(𝑃 − 1)) ∧ ∀𝑦 ∈ (1...(𝑃 − 1))((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1))))
28	7, 18, 27	sylanbrc 582	. . . 4 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ∈ 𝐴)
29		fzfi 14023	. . . . 5 ⊢ (1...(𝑃 − 1)) ∈ Fin
30		eleq1 2832	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
31		reseq2 6004	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ( I ↾ 𝑠) = ( I ↾ 𝑡))
32	31	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝑇 Σ_g ( I ↾ 𝑠)) = (𝑇 Σ_g ( I ↾ 𝑡)))
33	32	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃))
34	33	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃) ↔ ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃)))
35	30, 34	imbi12d 344	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ((𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ↔ (𝑡 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃))))
36	35	imbi2d 340	. . . . . 6 ⊢ (𝑠 = 𝑡 → ((𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃))) ↔ (𝑃 ∈ ℙ → (𝑡 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃)))))
37		eleq1 2832	. . . . . . . 8 ⊢ (𝑠 = (1...(𝑃 − 1)) → (𝑠 ∈ 𝐴 ↔ (1...(𝑃 − 1)) ∈ 𝐴))
38		reseq2 6004	. . . . . . . . . . 11 ⊢ (𝑠 = (1...(𝑃 − 1)) → ( I ↾ 𝑠) = ( I ↾ (1...(𝑃 − 1))))
39	38	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑠 = (1...(𝑃 − 1)) → (𝑇 Σ_g ( I ↾ 𝑠)) = (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))))
40	39	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑠 = (1...(𝑃 − 1)) → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃))
41	40	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑠 = (1...(𝑃 − 1)) → (((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃) ↔ ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃)))
42	37, 41	imbi12d 344	. . . . . . 7 ⊢ (𝑠 = (1...(𝑃 − 1)) → ((𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ↔ ((1...(𝑃 − 1)) ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃))))
43	42	imbi2d 340	. . . . . 6 ⊢ (𝑠 = (1...(𝑃 − 1)) → ((𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃))) ↔ (𝑃 ∈ ℙ → ((1...(𝑃 − 1)) ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃)))))
44		bi2.04 387	. . . . . . . . . . 11 ⊢ ((𝑠 ⊊ 𝑡 → (𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) ↔ (𝑃 ∈ ℙ → (𝑠 ⊊ 𝑡 → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))))
45		pm2.27 42	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → ((𝑃 ∈ ℙ → (𝑠 ⊊ 𝑡 → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → (𝑠 ⊊ 𝑡 → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))))
46	45	com34 91	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → ((𝑃 ∈ ℙ → (𝑠 ⊊ 𝑡 → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → (𝑠 ∈ 𝐴 → (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))))
47	44, 46	biimtrid 242	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → ((𝑠 ⊊ 𝑡 → (𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → (𝑠 ∈ 𝐴 → (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))))
48	47	alimdv 1915	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (∀𝑠(𝑠 ⊊ 𝑡 → (𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → ∀𝑠(𝑠 ∈ 𝐴 → (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))))
49		df-ral 3068	. . . . . . . . 9 ⊢ (∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ↔ ∀𝑠(𝑠 ∈ 𝐴 → (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃))))
50	48, 49	imbitrrdi 252	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (∀𝑠(𝑠 ⊊ 𝑡 → (𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃))))
51		wilthlem.t	. . . . . . . . . 10 ⊢ 𝑇 = (mulGrp‘ℂ_fld)
52		simp1 1136	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ∧ 𝑡 ∈ 𝐴) → 𝑃 ∈ ℙ)
53		simp3 1138	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ∧ 𝑡 ∈ 𝐴) → 𝑡 ∈ 𝐴)
54		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ∧ 𝑡 ∈ 𝐴) → ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))
55	51, 25, 52, 53, 54	wilthlem2 27130	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) ∧ 𝑡 ∈ 𝐴) → ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃))
56	55	3exp 1119	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (∀𝑠 ∈ 𝐴 (𝑠 ⊊ 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)) → (𝑡 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃))))
57	50, 56	syldc 48	. . . . . . 7 ⊢ (∀𝑠(𝑠 ⊊ 𝑡 → (𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → (𝑃 ∈ ℙ → (𝑡 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃))))
58	57	a1i 11	. . . . . 6 ⊢ (𝑡 ∈ Fin → (∀𝑠(𝑠 ⊊ 𝑡 → (𝑃 ∈ ℙ → (𝑠 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = (-1 mod 𝑃)))) → (𝑃 ∈ ℙ → (𝑡 ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃) = (-1 mod 𝑃)))))
59	36, 43, 58	findcard3 9346	. . . . 5 ⊢ ((1...(𝑃 − 1)) ∈ Fin → (𝑃 ∈ ℙ → ((1...(𝑃 − 1)) ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃))))
60	29, 59	ax-mp 5	. . . 4 ⊢ (𝑃 ∈ ℙ → ((1...(𝑃 − 1)) ∈ 𝐴 → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃)))
61	28, 60	mpd 15	. . 3 ⊢ (𝑃 ∈ ℙ → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃))
62		cnfld1 21429	. . . . . 6 ⊢ 1 = (1_r‘ℂ_fld)
63	51, 62	ringidval 20210	. . . . 5 ⊢ 1 = (0_g‘𝑇)
64		cncrng 21424	. . . . . 6 ⊢ ℂ_fld ∈ CRing
65	51	crngmgp 20268	. . . . . 6 ⊢ (ℂ_fld ∈ CRing → 𝑇 ∈ CMnd)
66	64, 65	mp1i 13	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑇 ∈ CMnd)
67	29	a1i 11	. . . . 5 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ∈ Fin)
68		zsubrg 21461	. . . . . 6 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
69	51	subrgsubm 20613	. . . . . 6 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘𝑇))
70	68, 69	mp1i 13	. . . . 5 ⊢ (𝑃 ∈ ℙ → ℤ ∈ (SubMnd‘𝑇))
71		f1oi 6900	. . . . . . . 8 ⊢ ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))–1-1-onto→(1...(𝑃 − 1))
72		f1of 6862	. . . . . . . 8 ⊢ (( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))–1-1-onto→(1...(𝑃 − 1)) → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶(1...(𝑃 − 1)))
73	71, 72	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶(1...(𝑃 − 1))
74		fzssz 13586	. . . . . . 7 ⊢ (1...(𝑃 − 1)) ⊆ ℤ
75		fss 6763	. . . . . . 7 ⊢ ((( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶(1...(𝑃 − 1)) ∧ (1...(𝑃 − 1)) ⊆ ℤ) → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℤ)
76	73, 74, 75	mp2an 691	. . . . . 6 ⊢ ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℤ
77	76	a1i 11	. . . . 5 ⊢ (𝑃 ∈ ℙ → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℤ)
78		1ex 11286	. . . . . . 7 ⊢ 1 ∈ V
79	78	a1i 11	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ V)
80	77, 67, 79	fdmfifsupp 9444	. . . . 5 ⊢ (𝑃 ∈ ℙ → ( I ↾ (1...(𝑃 − 1))) finSupp 1)
81	63, 66, 67, 70, 77, 80	gsumsubmcl 19961	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) ∈ ℤ)
82		1z 12673	. . . . 5 ⊢ 1 ∈ ℤ
83		znegcl 12678	. . . . 5 ⊢ (1 ∈ ℤ → -1 ∈ ℤ)
84	82, 83	mp1i 13	. . . 4 ⊢ (𝑃 ∈ ℙ → -1 ∈ ℤ)
85		moddvds 16313	. . . 4 ⊢ ((𝑃 ∈ ℕ ∧ (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) ∈ ℤ ∧ -1 ∈ ℤ) → (((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1)))
86	11, 81, 84, 85	syl3anc 1371	. . 3 ⊢ (𝑃 ∈ ℙ → (((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃) = (-1 mod 𝑃) ↔ 𝑃 ∥ ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1)))
87	61, 86	mpbid 232	. 2 ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1))
88		fcoi1 6795	. . . . . . . . . 10 ⊢ (( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶(1...(𝑃 − 1)) → (( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))) = ( I ↾ (1...(𝑃 − 1))))
89	73, 88	ax-mp 5	. . . . . . . . 9 ⊢ (( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))) = ( I ↾ (1...(𝑃 − 1)))
90	89	fveq1i 6921	. . . . . . . 8 ⊢ ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1))))‘𝑘) = (( I ↾ (1...(𝑃 − 1)))‘𝑘)
91		fvres 6939	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(𝑃 − 1)) → (( I ↾ (1...(𝑃 − 1)))‘𝑘) = ( I ‘𝑘))
92	90, 91	eqtrid 2792	. . . . . . 7 ⊢ (𝑘 ∈ (1...(𝑃 − 1)) → ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1))))‘𝑘) = ( I ‘𝑘))
93	92	adantl 481	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (1...(𝑃 − 1))) → ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1))))‘𝑘) = ( I ‘𝑘))
94	5, 93	seqfveq 14077	. . . . 5 ⊢ (𝑃 ∈ ℙ → (seq1( · , (( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))))‘(𝑃 − 1)) = (seq1( · , I )‘(𝑃 − 1)))
95		cnfldbas 21391	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
96	51, 95	mgpbas 20167	. . . . . 6 ⊢ ℂ = (Base‘𝑇)
97		cnfldmul 21395	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
98	51, 97	mgpplusg 20165	. . . . . 6 ⊢ · = (+_g‘𝑇)
99		eqid 2740	. . . . . 6 ⊢ (Cntz‘𝑇) = (Cntz‘𝑇)
100		cnring 21426	. . . . . . 7 ⊢ ℂ_fld ∈ Ring
101	51	ringmgp 20266	. . . . . . 7 ⊢ (ℂ_fld ∈ Ring → 𝑇 ∈ Mnd)
102	100, 101	mp1i 13	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑇 ∈ Mnd)
103		zsscn 12647	. . . . . . . 8 ⊢ ℤ ⊆ ℂ
104		fss 6763	. . . . . . . 8 ⊢ ((( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℤ ∧ ℤ ⊆ ℂ) → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℂ)
105	76, 103, 104	mp2an 691	. . . . . . 7 ⊢ ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℂ
106	105	a1i 11	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℂ)
107	96, 99, 66, 106	cntzcmnf 19887	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ran ( I ↾ (1...(𝑃 − 1))) ⊆ ((Cntz‘𝑇)‘ran ( I ↾ (1...(𝑃 − 1)))))
108		f1of1 6861	. . . . . . 7 ⊢ (( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))–1-1-onto→(1...(𝑃 − 1)) → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))–1-1→(1...(𝑃 − 1)))
109	71, 108	mp1i 13	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))–1-1→(1...(𝑃 − 1)))
110		suppssdm 8218	. . . . . . . . 9 ⊢ (( I ↾ (1...(𝑃 − 1))) supp 1) ⊆ dom ( I ↾ (1...(𝑃 − 1)))
111		dmresi 6081	. . . . . . . . 9 ⊢ dom ( I ↾ (1...(𝑃 − 1))) = (1...(𝑃 − 1))
112	110, 111	sseqtri 4045	. . . . . . . 8 ⊢ (( I ↾ (1...(𝑃 − 1))) supp 1) ⊆ (1...(𝑃 − 1))
113		rnresi 6104	. . . . . . . 8 ⊢ ran ( I ↾ (1...(𝑃 − 1))) = (1...(𝑃 − 1))
114	112, 113	sseqtrri 4046	. . . . . . 7 ⊢ (( I ↾ (1...(𝑃 − 1))) supp 1) ⊆ ran ( I ↾ (1...(𝑃 − 1)))
115	114	a1i 11	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (( I ↾ (1...(𝑃 − 1))) supp 1) ⊆ ran ( I ↾ (1...(𝑃 − 1))))
116		eqid 2740	. . . . . 6 ⊢ ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))) supp 1) = ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))) supp 1)
117	96, 63, 98, 99, 102, 67, 106, 107, 3, 109, 115, 116	gsumval3 19949	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) = (seq1( · , (( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))))‘(𝑃 − 1)))
118		facnn 14324	. . . . . 6 ⊢ ((𝑃 − 1) ∈ ℕ → (!‘(𝑃 − 1)) = (seq1( · , I )‘(𝑃 − 1)))
119	3, 118	syl 17	. . . . 5 ⊢ (𝑃 ∈ ℙ → (!‘(𝑃 − 1)) = (seq1( · , I )‘(𝑃 − 1)))
120	94, 117, 119	3eqtr4d 2790	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) = (!‘(𝑃 − 1)))
121	120	oveq1d 7463	. . 3 ⊢ (𝑃 ∈ ℙ → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1) = ((!‘(𝑃 − 1)) − -1))
122		nnm1nn0 12594	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
123	11, 122	syl 17	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ ℕ₀)
124	123	faccld 14333	. . . . 5 ⊢ (𝑃 ∈ ℙ → (!‘(𝑃 − 1)) ∈ ℕ)
125	124	nncnd 12309	. . . 4 ⊢ (𝑃 ∈ ℙ → (!‘(𝑃 − 1)) ∈ ℂ)
126		ax-1cn 11242	. . . 4 ⊢ 1 ∈ ℂ
127		subneg 11585	. . . 4 ⊢ (((!‘(𝑃 − 1)) ∈ ℂ ∧ 1 ∈ ℂ) → ((!‘(𝑃 − 1)) − -1) = ((!‘(𝑃 − 1)) + 1))
128	125, 126, 127	sylancl 585	. . 3 ⊢ (𝑃 ∈ ℙ → ((!‘(𝑃 − 1)) − -1) = ((!‘(𝑃 − 1)) + 1))
129	121, 128	eqtrd 2780	. 2 ⊢ (𝑃 ∈ ℙ → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1) = ((!‘(𝑃 − 1)) + 1))
130	87, 129	breqtrd 5192	1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∀wal 1535 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 ⊆ wss 3976 ⊊ wpss 3977 𝒫 cpw 4622 class class class wbr 5166 I cid 5592 dom cdm 5700 ran crn 5701 ↾ cres 5702 ∘ ccom 5704 ⟶wf 6569 –1-1→wf1 6570 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 supp csupp 8201 Fincfn 9003 ℂcc 11182 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 -cneg 11521 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ...cfz 13567 mod cmo 13920 seqcseq 14052 ↑cexp 14112 !cfa 14322 ∥ cdvds 16302 ℙcprime 16718 Σ_g cgsu 17500 Mndcmnd 18772 SubMndcsubmnd 18817 Cntzccntz 19355 CMndccmn 19822 mulGrpcmgp 20161 Ringcrg 20260 CRingccrg 20261 SubRingcsubrg 20595 ℂ_fldccnfld 21387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-phi 16813 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-cnfld 21388

This theorem is referenced by: wilth 27132

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