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Theorem freshmansdream 21495

Description: For a prime number 𝑃, if 𝑋 and 𝑌 are members of a commutative ring 𝑅 of characteristic 𝑃, then ((𝑋 + 𝑌)↑𝑃) = ((𝑋↑𝑃) + (𝑌↑𝑃)). This theorem is sometimes referred to as "the freshman's dream" . (Contributed by Thierry Arnoux, 18-Sep-2023.)

**Hypotheses**
Ref	Expression
freshmansdream.s	⊢ 𝐵 = (Base‘𝑅)
freshmansdream.a	⊢ + = (+_g‘𝑅)
freshmansdream.p	⊢ ↑ = (._g‘(mulGrp‘𝑅))
freshmansdream.c	⊢ 𝑃 = (chr‘𝑅)
freshmansdream.r	⊢ (𝜑 → 𝑅 ∈ CRing)
freshmansdream.1	⊢ (𝜑 → 𝑃 ∈ ℙ)
freshmansdream.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
freshmansdream.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
freshmansdream	⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌)))

Proof of Theorem freshmansdream Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		freshmansdream.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2		crngring 20176	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		freshmansdream.c	. . . . 5 ⊢ 𝑃 = (chr‘𝑅)
4	3	chrcl 21441	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ ℕ₀)
5	1, 2, 4	3syl 18	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ₀)
6		freshmansdream.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		freshmansdream.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		freshmansdream.s	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
9		eqid 2727	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
10		eqid 2727	. . . 4 ⊢ (._g‘𝑅) = (._g‘𝑅)
11		freshmansdream.a	. . . 4 ⊢ + = (+_g‘𝑅)
12		eqid 2727	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
13		freshmansdream.p	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑅))
14	8, 9, 10, 11, 12, 13	crngbinom 20260	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ ℕ₀) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑃 ↑ (𝑋 + 𝑌)) = (𝑅 Σ_g (𝑖 ∈ (0...𝑃) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))))
15	1, 5, 6, 7, 14	syl22anc 838	. 2 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = (𝑅 Σ_g (𝑖 ∈ (0...𝑃) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))))
16	5	nn0cnd 12556	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
17		1cnd 11231	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
18	16, 17	npcand 11597	. . . . . 6 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
19	18	oveq2d 7430	. . . . 5 ⊢ (𝜑 → (0...((𝑃 − 1) + 1)) = (0...𝑃))
20	19	eqcomd 2733	. . . 4 ⊢ (𝜑 → (0...𝑃) = (0...((𝑃 − 1) + 1)))
21	20	mpteq1d 5237	. . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑃) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))) = (𝑖 ∈ (0...((𝑃 − 1) + 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))))
22	21	oveq2d 7430	. 2 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (0...𝑃) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (𝑅 Σ_g (𝑖 ∈ (0...((𝑃 − 1) + 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))))
23		ringcmn 20207	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
24	1, 2, 23	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
25		freshmansdream.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
26		prmnn 16636	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
27		nnm1nn0 12535	. . . . 5 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
28	25, 26, 27	3syl 18	. . . 4 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
29		ringgrp 20169	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
30	1, 2, 29	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑅 ∈ Grp)
32	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑃 ∈ ℕ₀)
33		fzssz 13527	. . . . . . . . 9 ⊢ (0...((𝑃 − 1) + 1)) ⊆ ℤ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑃 − 1) + 1)) ⊆ ℤ)
35	34	sselda 3978	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ ℤ)
36		bccl 14305	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ₀ ∧ 𝑖 ∈ ℤ) → (𝑃C𝑖) ∈ ℕ₀)
37	32, 35, 36	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑃C𝑖) ∈ ℕ₀)
38	37	nn0zd 12606	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑃C𝑖) ∈ ℤ)
39	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑅 ∈ Ring)
41	12, 8	mgpbas 20071	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
42	12	ringmgp 20170	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
43	39, 42	syl 17	. . . . . . . 8 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
44	43	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (mulGrp‘𝑅) ∈ Mnd)
45		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ (0...((𝑃 − 1) + 1)))
46	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (0...((𝑃 − 1) + 1)) = (0...𝑃))
47	45, 46	eleqtrd 2830	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ (0...𝑃))
48		fznn0sub 13557	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑃) → (𝑃 − 𝑖) ∈ ℕ₀)
49	47, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑃 − 𝑖) ∈ ℕ₀)
50	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑋 ∈ 𝐵)
51	41, 13, 44, 49, 50	mulgnn0cld 19041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵)
52		elfznn0 13618	. . . . . . . 8 ⊢ (𝑖 ∈ (0...((𝑃 − 1) + 1)) → 𝑖 ∈ ℕ₀)
53	52	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ ℕ₀)
54	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑌 ∈ 𝐵)
55	41, 13, 44, 53, 54	mulgnn0cld 19041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑖 ↑ 𝑌) ∈ 𝐵)
56	8, 9	ringcl 20181	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵 ∧ (𝑖 ↑ 𝑌) ∈ 𝐵) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
57	40, 51, 55, 56	syl3anc 1369	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
58	8, 10	mulgcl 19037	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑃C𝑖) ∈ ℤ ∧ (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) ∈ 𝐵)
59	31, 38, 57, 58	syl3anc 1369	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) ∈ 𝐵)
60	8, 11, 24, 28, 59	gsummptfzsplit 19878	. . 3 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (0...((𝑃 − 1) + 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = ((𝑅 Σ_g (𝑖 ∈ (0...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) + (𝑅 Σ_g (𝑖 ∈ {((𝑃 − 1) + 1)} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))))))
61	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑅 ∈ Grp)
62		elfzelz 13525	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...(𝑃 − 1)) → 𝑖 ∈ ℤ)
63	5, 62, 36	syl2an 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑃C𝑖) ∈ ℕ₀)
64	63	nn0zd 12606	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑃C𝑖) ∈ ℤ)
65	39	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑅 ∈ Ring)
66	65, 42	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (mulGrp‘𝑅) ∈ Mnd)
67		fzssp1 13568	. . . . . . . . . . . 12 ⊢ (0...(𝑃 − 1)) ⊆ (0...((𝑃 − 1) + 1))
68	67, 19	sseqtrid 4030	. . . . . . . . . . 11 ⊢ (𝜑 → (0...(𝑃 − 1)) ⊆ (0...𝑃))
69	68	sselda 3978	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑖 ∈ (0...𝑃))
70	69, 48	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑃 − 𝑖) ∈ ℕ₀)
71	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑋 ∈ 𝐵)
72	41, 13, 66, 70, 71	mulgnn0cld 19041	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵)
73		elfznn0 13618	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...(𝑃 − 1)) → 𝑖 ∈ ℕ₀)
74	73	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑖 ∈ ℕ₀)
75	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑌 ∈ 𝐵)
76	41, 13, 66, 74, 75	mulgnn0cld 19041	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑖 ↑ 𝑌) ∈ 𝐵)
77	65, 72, 76, 56	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
78	61, 64, 77, 58	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) ∈ 𝐵)
79	8, 11, 24, 28, 78	gsummptfzsplitl 19879	. . . . 5 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (0...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = ((𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) + (𝑅 Σ_g (𝑖 ∈ {0} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))))))
80	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑅 ∈ Ring)
81		prmdvdsbc 16689	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑖))
82	25, 81	sylan 579	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑖))
83	80, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (mulGrp‘𝑅) ∈ Mnd)
84	5	nn0zd 12606	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
85		1nn0 12510	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
86		eluzmn 12851	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
87	84, 85, 86	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
88		fzss2 13565	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘(𝑃 − 1)) → (1...(𝑃 − 1)) ⊆ (1...𝑃))
89	87, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...(𝑃 − 1)) ⊆ (1...𝑃))
90	89	sselda 3978	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑖 ∈ (1...𝑃))
91		fznn0sub 13557	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑃) → (𝑃 − 𝑖) ∈ ℕ₀)
92	90, 91	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (𝑃 − 𝑖) ∈ ℕ₀)
93	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑋 ∈ 𝐵)
94	41, 13, 83, 92, 93	mulgnn0cld 19041	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵)
95		elfznn 13554	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...(𝑃 − 1)) → 𝑖 ∈ ℕ)
96	95	nnnn0d 12554	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...(𝑃 − 1)) → 𝑖 ∈ ℕ₀)
97	96	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑖 ∈ ℕ₀)
98	7	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑌 ∈ 𝐵)
99	41, 13, 83, 97, 98	mulgnn0cld 19041	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (𝑖 ↑ 𝑌) ∈ 𝐵)
100	80, 94, 99, 56	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
101		eqid 2727	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
102	3, 8, 10, 101	dvdschrmulg 21445	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∥ (𝑃C𝑖) ∧ (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = (0_g‘𝑅))
103	80, 82, 100, 102	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = (0_g‘𝑅))
104	103	mpteq2dva 5242	. . . . . . . 8 ⊢ (𝜑 → (𝑖 ∈ (1...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))) = (𝑖 ∈ (1...(𝑃 − 1)) ↦ (0_g‘𝑅)))
105	104	oveq2d 7430	. . . . . . 7 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ (0_g‘𝑅))))
106		ringmnd 20174	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
107	39, 106	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
108		ovex 7447	. . . . . . . 8 ⊢ (1...(𝑃 − 1)) ∈ V
109	101	gsumz 18779	. . . . . . . 8 ⊢ ((𝑅 ∈ Mnd ∧ (1...(𝑃 − 1)) ∈ V) → (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ (0_g‘𝑅))) = (0_g‘𝑅))
110	107, 108, 109	sylancl 585	. . . . . . 7 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ (0_g‘𝑅))) = (0_g‘𝑅))
111	105, 110	eqtrd 2767	. . . . . 6 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (0_g‘𝑅))
112		0nn0 12509	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
113	112	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
114	41, 13, 43, 5, 6	mulgnn0cld 19041	. . . . . . 7 ⊢ (𝜑 → (𝑃 ↑ 𝑋) ∈ 𝐵)
115		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0)
116	115	oveq2d 7430	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑃C𝑖) = (𝑃C0))
117	115	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑃 − 𝑖) = (𝑃 − 0))
118	117	oveq1d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃 − 𝑖) ↑ 𝑋) = ((𝑃 − 0) ↑ 𝑋))
119	115	oveq1d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑖 ↑ 𝑌) = (0 ↑ 𝑌))
120	118, 119	oveq12d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) = (((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌)))
121	116, 120	oveq12d 7432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))))
122		bcn0 14293	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ₀ → (𝑃C0) = 1)
123	5, 122	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃C0) = 1)
124	16	subid1d 11582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 0) = 𝑃)
125	124	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 − 0) ↑ 𝑋) = (𝑃 ↑ 𝑋))
126		eqid 2727	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
127	12, 126	ringidval 20114	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑅) = (0_g‘(mulGrp‘𝑅))
128	41, 127, 13	mulg0 19021	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ 𝐵 → (0 ↑ 𝑌) = (1_r‘𝑅))
129	7, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ↑ 𝑌) = (1_r‘𝑅))
130	125, 129	oveq12d 7432	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌)) = ((𝑃 ↑ 𝑋)(._r‘𝑅)(1_r‘𝑅)))
131	8, 9, 126	ringridm 20195	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑃 ↑ 𝑋) ∈ 𝐵) → ((𝑃 ↑ 𝑋)(._r‘𝑅)(1_r‘𝑅)) = (𝑃 ↑ 𝑋))
132	39, 114, 131	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 ↑ 𝑋)(._r‘𝑅)(1_r‘𝑅)) = (𝑃 ↑ 𝑋))
133	130, 132	eqtrd 2767	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌)) = (𝑃 ↑ 𝑋))
134	123, 133	oveq12d 7432	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))) = (1(._g‘𝑅)(𝑃 ↑ 𝑋)))
135	8, 10	mulg1 19027	. . . . . . . . . . 11 ⊢ ((𝑃 ↑ 𝑋) ∈ 𝐵 → (1(._g‘𝑅)(𝑃 ↑ 𝑋)) = (𝑃 ↑ 𝑋))
136	114, 135	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1(._g‘𝑅)(𝑃 ↑ 𝑋)) = (𝑃 ↑ 𝑋))
137	134, 136	eqtrd 2767	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))) = (𝑃 ↑ 𝑋))
138	137	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))) = (𝑃 ↑ 𝑋))
139	121, 138	eqtrd 2767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = (𝑃 ↑ 𝑋))
140	8, 107, 113, 114, 139	gsumsnd 19898	. . . . . 6 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ {0} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (𝑃 ↑ 𝑋))
141	111, 140	oveq12d 7432	. . . . 5 ⊢ (𝜑 → ((𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) + (𝑅 Σ_g (𝑖 ∈ {0} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))))) = ((0_g‘𝑅) + (𝑃 ↑ 𝑋)))
142	8, 11, 101	grplid 18915	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝑃 ↑ 𝑋) ∈ 𝐵) → ((0_g‘𝑅) + (𝑃 ↑ 𝑋)) = (𝑃 ↑ 𝑋))
143	30, 114, 142	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((0_g‘𝑅) + (𝑃 ↑ 𝑋)) = (𝑃 ↑ 𝑋))
144	79, 141, 143	3eqtrd 2771	. . . 4 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (0...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (𝑃 ↑ 𝑋))
145	18, 5	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → ((𝑃 − 1) + 1) ∈ ℕ₀)
146	41, 13, 43, 5, 7	mulgnn0cld 19041	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝑌) ∈ 𝐵)
147		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → 𝑖 = ((𝑃 − 1) + 1))
148	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃 − 1) + 1) = 𝑃)
149	147, 148	eqtrd 2767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → 𝑖 = 𝑃)
150	149	oveq2d 7430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (𝑃C𝑖) = (𝑃C𝑃))
151	149	oveq2d 7430	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (𝑃 − 𝑖) = (𝑃 − 𝑃))
152	151	oveq1d 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃 − 𝑖) ↑ 𝑋) = ((𝑃 − 𝑃) ↑ 𝑋))
153	149	oveq1d 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (𝑖 ↑ 𝑌) = (𝑃 ↑ 𝑌))
154	152, 153	oveq12d 7432	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) = (((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌)))
155	150, 154	oveq12d 7432	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))))
156		bcnn 14295	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ₀ → (𝑃C𝑃) = 1)
157	5, 156	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑃C𝑃) = 1)
158	16	subidd 11581	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 𝑃) = 0)
159	158	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 − 𝑃) ↑ 𝑋) = (0 ↑ 𝑋))
160	41, 127, 13	mulg0 19021	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐵 → (0 ↑ 𝑋) = (1_r‘𝑅))
161	6, 160	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ↑ 𝑋) = (1_r‘𝑅))
162	159, 161	eqtrd 2767	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 − 𝑃) ↑ 𝑋) = (1_r‘𝑅))
163	162	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌)) = ((1_r‘𝑅)(._r‘𝑅)(𝑃 ↑ 𝑌)))
164	8, 9, 126	ringlidm 20194	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑃 ↑ 𝑌) ∈ 𝐵) → ((1_r‘𝑅)(._r‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
165	39, 146, 164	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((1_r‘𝑅)(._r‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
166	163, 165	eqtrd 2767	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
167	157, 166	oveq12d 7432	. . . . . . . 8 ⊢ (𝜑 → ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))) = (1(._g‘𝑅)(𝑃 ↑ 𝑌)))
168	8, 10	mulg1 19027	. . . . . . . . 9 ⊢ ((𝑃 ↑ 𝑌) ∈ 𝐵 → (1(._g‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
169	146, 168	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1(._g‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
170	167, 169	eqtrd 2767	. . . . . . 7 ⊢ (𝜑 → ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))) = (𝑃 ↑ 𝑌))
171	170	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))) = (𝑃 ↑ 𝑌))
172	155, 171	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = (𝑃 ↑ 𝑌))
173	8, 107, 145, 146, 172	gsumsnd 19898	. . . 4 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ {((𝑃 − 1) + 1)} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (𝑃 ↑ 𝑌))
174	144, 173	oveq12d 7432	. . 3 ⊢ (𝜑 → ((𝑅 Σ_g (𝑖 ∈ (0...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) + (𝑅 Σ_g (𝑖 ∈ {((𝑃 − 1) + 1)} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)))))) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌)))
175	60, 174	eqtrd 2767	. 2 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (0...((𝑃 − 1) + 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌)))
176	15, 22, 175	3eqtrd 2771	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ⊆ wss 3944 {csn 4624 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 0cc0 11130 1c1 11131 + caddc 11133 − cmin 11466 ℕcn 12234 ℕ₀cn0 12494 ℤcz 12580 ℤ_≥cuz 12844 ...cfz 13508 Ccbc 14285 ∥ cdvds 16222 ℙcprime 16633 Basecbs 17171 +_gcplusg 17224 ._rcmulr 17225 0_gc0g 17412 Σ_g cgsu 17413 Mndcmnd 18685 Grpcgrp 18881 ._gcmg 19014 CMndccmn 19726 mulGrpcmgp 20065 1_rcur 20112 Ringcrg 20164 CRingccrg 20165 chrcchr 21414

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-dvds 16223 df-gcd 16461 df-prm 16634 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-0g 17414 df-gsum 17415 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-cntz 19259 df-od 19474 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-chr 21418

This theorem is referenced by: ply1fermltlchr 22218 frobrhm 32918

W3C validator