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Theorem wilthlem3 26581

Description: Lemma for wilth 26582. Here we round out the argument of wilthlem2 26580 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 16635	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
2		uz2m1nn 12909	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (𝑃 − 1) ∈ ℕ)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ ℕ)
4		nnuz 12867	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
5	3, 4	eleqtrdi 2843	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ (ℤ_≥‘1))
6		eluzfz2 13511	. . . . . 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 483	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
9		elfzelz 13503	. . . . . . . . 9 ⊢ (𝑦 ∈ (1...(𝑃 − 1)) → 𝑦 ∈ ℤ)
10	9	adantl 482	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → 𝑦 ∈ ℤ)
11		prmnn 16613	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
12		fzm1ndvds 16267	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑦)
13	11, 12	sylan 580	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ 𝑦)
14		eqid 2732	. . . . . . . . 9 ⊢ ((𝑦↑(𝑃 − 2)) mod 𝑃) = ((𝑦↑(𝑃 − 2)) mod 𝑃)
15	14	prmdiv 16720	. . . . . . . 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 495	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑦 ∈ (1...(𝑃 − 1))) → ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)))
18	17	ralrimiva 3146	. . . . 5 ⊢ (𝑃 ∈ ℙ → ∀𝑦 ∈ (1...(𝑃 − 1))((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1)))
19		ovex 7444	. . . . . . 7 ⊢ (1...(𝑃 − 1)) ∈ V
20	19	pwid 4624	. . . . . 6 ⊢ (1...(𝑃 − 1)) ∈ 𝒫 (1...(𝑃 − 1))
21		eleq2 2822	. . . . . . . 8 ⊢ (𝑥 = (1...(𝑃 − 1)) → ((𝑃 − 1) ∈ 𝑥 ↔ (𝑃 − 1) ∈ (1...(𝑃 − 1))))
22		eleq2 2822	. . . . . . . . 9 ⊢ (𝑥 = (1...(𝑃 − 1)) → (((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ 𝑥 ↔ ((𝑦↑(𝑃 − 2)) mod 𝑃) ∈ (1...(𝑃 − 1))))
23	22	raleqbi1dv 3333	. . . . . . . 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 3686	. . . . . 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 707	. . . . 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 583	. . . 4 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ∈ 𝐴)
29		fzfi 13939	. . . . 5 ⊢ (1...(𝑃 − 1)) ∈ Fin
30		eleq1 2821	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑠 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
31		reseq2 5976	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ( I ↾ 𝑠) = ( I ↾ 𝑡))
32	31	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝑇 Σ_g ( I ↾ 𝑠)) = (𝑇 Σ_g ( I ↾ 𝑡)))
33	32	oveq1d 7426	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = ((𝑇 Σ_g ( I ↾ 𝑡)) mod 𝑃))
34	33	eqeq1d 2734	. . . . . . . 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 2821	. . . . . . . 8 ⊢ (𝑠 = (1...(𝑃 − 1)) → (𝑠 ∈ 𝐴 ↔ (1...(𝑃 − 1)) ∈ 𝐴))
38		reseq2 5976	. . . . . . . . . . 11 ⊢ (𝑠 = (1...(𝑃 − 1)) → ( I ↾ 𝑠) = ( I ↾ (1...(𝑃 − 1))))
39	38	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝑠 = (1...(𝑃 − 1)) → (𝑇 Σ_g ( I ↾ 𝑠)) = (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))))
40	39	oveq1d 7426	. . . . . . . . 9 ⊢ (𝑠 = (1...(𝑃 − 1)) → ((𝑇 Σ_g ( I ↾ 𝑠)) mod 𝑃) = ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) mod 𝑃))
41	40	eqeq1d 2734	. . . . . . . 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		bi2.04 388	. . . . . . . . . . 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 241	. . . . . . . . . 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 1919	. . . . . . . . 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 3062	. . . . . . . . 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 251	. . . . . . . 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 26580	. . . . . . . . 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 9287	. . . . 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 20976	. . . . . 6 ⊢ 1 = (1_r‘ℂ_fld)
63	51, 62	ringidval 20008	. . . . 5 ⊢ 1 = (0_g‘𝑇)
64		cncrng 20972	. . . . . 6 ⊢ ℂ_fld ∈ CRing
65	51	crngmgp 20066	. . . . . 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 21004	. . . . . 6 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
69	51	subrgsubm 20336	. . . . . 6 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubMnd‘𝑇))
70	68, 69	mp1i 13	. . . . 5 ⊢ (𝑃 ∈ ℙ → ℤ ∈ (SubMnd‘𝑇))
71		f1oi 6871	. . . . . . . 8 ⊢ ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))–1-1-onto→(1...(𝑃 − 1))
72		f1of 6833	. . . . . . . 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 13505	. . . . . . 7 ⊢ (1...(𝑃 − 1)) ⊆ ℤ
75		fss 6734	. . . . . . 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 690	. . . . . 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 11212	. . . . . . 7 ⊢ 1 ∈ V
79	78	a1i 11	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 1 ∈ V)
80	77, 67, 79	fdmfifsupp 9375	. . . . 5 ⊢ (𝑃 ∈ ℙ → ( I ↾ (1...(𝑃 − 1))) finSupp 1)
81	63, 66, 67, 70, 77, 80	gsumsubmcl 19789	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) ∈ ℤ)
82		1z 12594	. . . . 5 ⊢ 1 ∈ ℤ
83		znegcl 12599	. . . . 5 ⊢ (1 ∈ ℤ → -1 ∈ ℤ)
84	82, 83	mp1i 13	. . . 4 ⊢ (𝑃 ∈ ℙ → -1 ∈ ℤ)
85		moddvds 16210	. . . 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 231	. 2 ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1))
88		fcoi1 6765	. . . . . . . . . 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 6892	. . . . . . . 8 ⊢ ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1))))‘𝑘) = (( I ↾ (1...(𝑃 − 1)))‘𝑘)
91		fvres 6910	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(𝑃 − 1)) → (( I ↾ (1...(𝑃 − 1)))‘𝑘) = ( I ‘𝑘))
92	90, 91	eqtrid 2784	. . . . . . 7 ⊢ (𝑘 ∈ (1...(𝑃 − 1)) → ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1))))‘𝑘) = ( I ‘𝑘))
93	92	adantl 482	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑘 ∈ (1...(𝑃 − 1))) → ((( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1))))‘𝑘) = ( I ‘𝑘))
94	5, 93	seqfveq 13994	. . . . 5 ⊢ (𝑃 ∈ ℙ → (seq1( · , (( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))))‘(𝑃 − 1)) = (seq1( · , I )‘(𝑃 − 1)))
95		cnfldbas 20954	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
96	51, 95	mgpbas 19995	. . . . . 6 ⊢ ℂ = (Base‘𝑇)
97		cnfldmul 20956	. . . . . . 7 ⊢ · = (._r‘ℂ_fld)
98	51, 97	mgpplusg 19993	. . . . . 6 ⊢ · = (+_g‘𝑇)
99		eqid 2732	. . . . . 6 ⊢ (Cntz‘𝑇) = (Cntz‘𝑇)
100		cnring 20973	. . . . . . 7 ⊢ ℂ_fld ∈ Ring
101	51	ringmgp 20064	. . . . . . 7 ⊢ (ℂ_fld ∈ Ring → 𝑇 ∈ Mnd)
102	100, 101	mp1i 13	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑇 ∈ Mnd)
103		zsscn 12568	. . . . . . . 8 ⊢ ℤ ⊆ ℂ
104		fss 6734	. . . . . . . 8 ⊢ ((( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℤ ∧ ℤ ⊆ ℂ) → ( I ↾ (1...(𝑃 − 1))):(1...(𝑃 − 1))⟶ℂ)
105	76, 103, 104	mp2an 690	. . . . . . 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 19715	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ran ( I ↾ (1...(𝑃 − 1))) ⊆ ((Cntz‘𝑇)‘ran ( I ↾ (1...(𝑃 − 1)))))
108		f1of1 6832	. . . . . . 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 8164	. . . . . . . . 9 ⊢ (( I ↾ (1...(𝑃 − 1))) supp 1) ⊆ dom ( I ↾ (1...(𝑃 − 1)))
111		dmresi 6051	. . . . . . . . 9 ⊢ dom ( I ↾ (1...(𝑃 − 1))) = (1...(𝑃 − 1))
112	110, 111	sseqtri 4018	. . . . . . . 8 ⊢ (( I ↾ (1...(𝑃 − 1))) supp 1) ⊆ (1...(𝑃 − 1))
113		rnresi 6074	. . . . . . . 8 ⊢ ran ( I ↾ (1...(𝑃 − 1))) = (1...(𝑃 − 1))
114	112, 113	sseqtrri 4019	. . . . . . 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 2732	. . . . . 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 19777	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) = (seq1( · , (( I ↾ (1...(𝑃 − 1))) ∘ ( I ↾ (1...(𝑃 − 1)))))‘(𝑃 − 1)))
118		facnn 14237	. . . . . 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 2782	. . . 4 ⊢ (𝑃 ∈ ℙ → (𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) = (!‘(𝑃 − 1)))
121	120	oveq1d 7426	. . 3 ⊢ (𝑃 ∈ ℙ → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1) = ((!‘(𝑃 − 1)) − -1))
122		nnm1nn0 12515	. . . . . . 7 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
123	11, 122	syl 17	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ ℕ₀)
124	123	faccld 14246	. . . . 5 ⊢ (𝑃 ∈ ℙ → (!‘(𝑃 − 1)) ∈ ℕ)
125	124	nncnd 12230	. . . 4 ⊢ (𝑃 ∈ ℙ → (!‘(𝑃 − 1)) ∈ ℂ)
126		ax-1cn 11170	. . . 4 ⊢ 1 ∈ ℂ
127		subneg 11511	. . . 4 ⊢ (((!‘(𝑃 − 1)) ∈ ℂ ∧ 1 ∈ ℂ) → ((!‘(𝑃 − 1)) − -1) = ((!‘(𝑃 − 1)) + 1))
128	125, 126, 127	sylancl 586	. . 3 ⊢ (𝑃 ∈ ℙ → ((!‘(𝑃 − 1)) − -1) = ((!‘(𝑃 − 1)) + 1))
129	121, 128	eqtrd 2772	. 2 ⊢ (𝑃 ∈ ℙ → ((𝑇 Σ_g ( I ↾ (1...(𝑃 − 1)))) − -1) = ((!‘(𝑃 − 1)) + 1))
130	87, 129	breqtrd 5174	1 ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ ((!‘(𝑃 − 1)) + 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 ⊆ wss 3948 ⊊ wpss 3949 𝒫 cpw 4602 class class class wbr 5148 I cid 5573 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 supp csupp 8148 Fincfn 8941 ℂcc 11110 1c1 11113 + caddc 11115 · cmul 11117 − cmin 11446 -cneg 11447 ℕcn 12214 2c2 12269 ℕ₀cn0 12474 ℤcz 12560 ℤ_≥cuz 12824 ...cfz 13486 mod cmo 13836 seqcseq 13968 ↑cexp 14029 !cfa 14235 ∥ cdvds 16199 ℙcprime 16610 Σ_g cgsu 17388 Mndcmnd 18627 SubMndcsubmnd 18672 Cntzccntz 19181 CMndccmn 19650 mulGrpcmgp 19989 Ringcrg 20058 CRingccrg 20059 SubRingcsubrg 20319 ℂ_fldccnfld 20950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-dec 12680 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16200 df-gcd 16438 df-prm 16611 df-phi 16701 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-0g 17389 df-gsum 17390 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-mulg 18953 df-subg 19005 df-cntz 19183 df-cmn 19652 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-subrg 20321 df-cnfld 20951

This theorem is referenced by: wilth 26582

W3C validator