freshmansdream - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > freshmansdream		Structured version Visualization version GIF version

Theorem freshmansdream 21512

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 20189	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		freshmansdream.c	. . . . 5 ⊢ 𝑃 = (chr‘𝑅)
4	3	chrcl 21458	. . . 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 2725	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
10		eqid 2725	. . . 4 ⊢ (._g‘𝑅) = (._g‘𝑅)
11		freshmansdream.a	. . . 4 ⊢ + = (+_g‘𝑅)
12		eqid 2725	. . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
13		freshmansdream.p	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝑅))
14	8, 9, 10, 11, 12, 13	crngbinom 20275	. . 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 837	. 2 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = (𝑅 Σ_g (𝑖 ∈ (0...𝑃) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))))
16	5	nn0cnd 12564	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℂ)
17		1cnd 11239	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
18	16, 17	npcand 11605	. . . . . 6 ⊢ (𝜑 → ((𝑃 − 1) + 1) = 𝑃)
19	18	oveq2d 7433	. . . . 5 ⊢ (𝜑 → (0...((𝑃 − 1) + 1)) = (0...𝑃))
20	19	eqcomd 2731	. . . 4 ⊢ (𝜑 → (0...𝑃) = (0...((𝑃 − 1) + 1)))
21	20	mpteq1d 5243	. . 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 7433	. 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 20222	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
24	1, 2, 23	3syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
25		freshmansdream.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
26		prmnn 16644	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
27		nnm1nn0 12543	. . . . 5 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
28	25, 26, 27	3syl 18	. . . 4 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ₀)
29		ringgrp 20182	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
30	1, 2, 29	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
31	30	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑅 ∈ Grp)
32	5	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑃 ∈ ℕ₀)
33		fzssz 13535	. . . . . . . . 9 ⊢ (0...((𝑃 − 1) + 1)) ⊆ ℤ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0...((𝑃 − 1) + 1)) ⊆ ℤ)
35	34	sselda 3977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ ℤ)
36		bccl 14313	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ₀ ∧ 𝑖 ∈ ℤ) → (𝑃C𝑖) ∈ ℕ₀)
37	32, 35, 36	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑃C𝑖) ∈ ℕ₀)
38	37	nn0zd 12614	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑃C𝑖) ∈ ℤ)
39	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
40	39	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑅 ∈ Ring)
41	12, 8	mgpbas 20084	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
42	12	ringmgp 20183	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd)
43	39, 42	syl 17	. . . . . . . 8 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd)
44	43	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (mulGrp‘𝑅) ∈ Mnd)
45		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ (0...((𝑃 − 1) + 1)))
46	19	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (0...((𝑃 − 1) + 1)) = (0...𝑃))
47	45, 46	eleqtrd 2827	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ (0...𝑃))
48		fznn0sub 13565	. . . . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑋 ∈ 𝐵)
51	41, 13, 44, 49, 50	mulgnn0cld 19054	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵)
52		elfznn0 13626	. . . . . . . 8 ⊢ (𝑖 ∈ (0...((𝑃 − 1) + 1)) → 𝑖 ∈ ℕ₀)
53	52	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑖 ∈ ℕ₀)
54	7	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → 𝑌 ∈ 𝐵)
55	41, 13, 44, 53, 54	mulgnn0cld 19054	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (𝑖 ↑ 𝑌) ∈ 𝐵)
56	8, 9	ringcl 20194	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵 ∧ (𝑖 ↑ 𝑌) ∈ 𝐵) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
57	40, 51, 55, 56	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...((𝑃 − 1) + 1))) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
58	8, 10	mulgcl 19050	. . . . 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 1368	. . . 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 19891	. . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑅 ∈ Grp)
62		elfzelz 13533	. . . . . . . . 9 ⊢ (𝑖 ∈ (0...(𝑃 − 1)) → 𝑖 ∈ ℤ)
63	5, 62, 36	syl2an 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑃C𝑖) ∈ ℕ₀)
64	63	nn0zd 12614	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑃C𝑖) ∈ ℤ)
65	39	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑅 ∈ Ring)
66	65, 42	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (mulGrp‘𝑅) ∈ Mnd)
67		fzssp1 13576	. . . . . . . . . . . 12 ⊢ (0...(𝑃 − 1)) ⊆ (0...((𝑃 − 1) + 1))
68	67, 19	sseqtrid 4030	. . . . . . . . . . 11 ⊢ (𝜑 → (0...(𝑃 − 1)) ⊆ (0...𝑃))
69	68	sselda 3977	. . . . . . . . . 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 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑋 ∈ 𝐵)
72	41, 13, 66, 70, 71	mulgnn0cld 19054	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵)
73		elfznn0 13626	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0...(𝑃 − 1)) → 𝑖 ∈ ℕ₀)
74	73	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑖 ∈ ℕ₀)
75	7	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → 𝑌 ∈ 𝐵)
76	41, 13, 66, 74, 75	mulgnn0cld 19054	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (𝑖 ↑ 𝑌) ∈ 𝐵)
77	65, 72, 76, 56	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
78	61, 64, 77, 58	syl3anc 1368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑃 − 1))) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) ∈ 𝐵)
79	8, 11, 24, 28, 78	gsummptfzsplitl 19892	. . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑅 ∈ Ring)
81		prmdvdsbc 16697	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑖))
82	25, 81	sylan 578	. . . . . . . . . 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 12614	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
85		1nn0 12518	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
86		eluzmn 12859	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
87	84, 85, 86	sylancl 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
88		fzss2 13573	. . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑖 ∈ (1...𝑃))
91		fznn0sub 13565	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...𝑃) → (𝑃 − 𝑖) ∈ ℕ₀)
92	90, 91	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (𝑃 − 𝑖) ∈ ℕ₀)
93	6	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑋 ∈ 𝐵)
94	41, 13, 83, 92, 93	mulgnn0cld 19054	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → ((𝑃 − 𝑖) ↑ 𝑋) ∈ 𝐵)
95		elfznn 13562	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...(𝑃 − 1)) → 𝑖 ∈ ℕ)
96	95	nnnn0d 12562	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (1...(𝑃 − 1)) → 𝑖 ∈ ℕ₀)
97	96	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑖 ∈ ℕ₀)
98	7	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → 𝑌 ∈ 𝐵)
99	41, 13, 83, 97, 98	mulgnn0cld 19054	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (𝑖 ↑ 𝑌) ∈ 𝐵)
100	80, 94, 99, 56	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) ∈ 𝐵)
101		eqid 2725	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
102	3, 8, 10, 101	dvdschrmulg 21462	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑃 − 1))) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = (0_g‘𝑅))
104	103	mpteq2dva 5248	. . . . . . . 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 7433	. . . . . . 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 20187	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
107	39, 106	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Mnd)
108		ovex 7450	. . . . . . . 8 ⊢ (1...(𝑃 − 1)) ∈ V
109	101	gsumz 18792	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ (0_g‘𝑅))) = (0_g‘𝑅))
111	105, 110	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ (1...(𝑃 − 1)) ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (0_g‘𝑅))
112		0nn0 12517	. . . . . . . 8 ⊢ 0 ∈ ℕ₀
113	112	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℕ₀)
114	41, 13, 43, 5, 6	mulgnn0cld 19054	. . . . . . 7 ⊢ (𝜑 → (𝑃 ↑ 𝑋) ∈ 𝐵)
115		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → 𝑖 = 0)
116	115	oveq2d 7433	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑃C𝑖) = (𝑃C0))
117	115	oveq2d 7433	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑃 − 𝑖) = (𝑃 − 0))
118	117	oveq1d 7432	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃 − 𝑖) ↑ 𝑋) = ((𝑃 − 0) ↑ 𝑋))
119	115	oveq1d 7432	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 = 0) → (𝑖 ↑ 𝑌) = (0 ↑ 𝑌))
120	118, 119	oveq12d 7435	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = 0) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) = (((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌)))
121	116, 120	oveq12d 7435	. . . . . . . 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 14301	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℕ₀ → (𝑃C0) = 1)
123	5, 122	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃C0) = 1)
124	16	subid1d 11590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑃 − 0) = 𝑃)
125	124	oveq1d 7432	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑃 − 0) ↑ 𝑋) = (𝑃 ↑ 𝑋))
126		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
127	12, 126	ringidval 20127	. . . . . . . . . . . . . . 15 ⊢ (1_r‘𝑅) = (0_g‘(mulGrp‘𝑅))
128	41, 127, 13	mulg0 19034	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ 𝐵 → (0 ↑ 𝑌) = (1_r‘𝑅))
129	7, 128	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ↑ 𝑌) = (1_r‘𝑅))
130	125, 129	oveq12d 7435	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌)) = ((𝑃 ↑ 𝑋)(._r‘𝑅)(1_r‘𝑅)))
131	8, 9, 126	ringridm 20210	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑃 ↑ 𝑋) ∈ 𝐵) → ((𝑃 ↑ 𝑋)(._r‘𝑅)(1_r‘𝑅)) = (𝑃 ↑ 𝑋))
132	39, 114, 131	syl2anc 582	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 ↑ 𝑋)(._r‘𝑅)(1_r‘𝑅)) = (𝑃 ↑ 𝑋))
133	130, 132	eqtrd 2765	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌)) = (𝑃 ↑ 𝑋))
134	123, 133	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))) = (1(._g‘𝑅)(𝑃 ↑ 𝑋)))
135	8, 10	mulg1 19040	. . . . . . . . . . 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 2765	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))) = (𝑃 ↑ 𝑋))
138	137	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃C0)(._g‘𝑅)(((𝑃 − 0) ↑ 𝑋)(._r‘𝑅)(0 ↑ 𝑌))) = (𝑃 ↑ 𝑋))
139	121, 138	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))) = (𝑃 ↑ 𝑋))
140	8, 107, 113, 114, 139	gsumsnd 19911	. . . . . 6 ⊢ (𝜑 → (𝑅 Σ_g (𝑖 ∈ {0} ↦ ((𝑃C𝑖)(._g‘𝑅)(((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌))))) = (𝑃 ↑ 𝑋))
141	111, 140	oveq12d 7435	. . . . 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 18928	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝑃 ↑ 𝑋) ∈ 𝐵) → ((0_g‘𝑅) + (𝑃 ↑ 𝑋)) = (𝑃 ↑ 𝑋))
143	30, 114, 142	syl2anc 582	. . . . 5 ⊢ (𝜑 → ((0_g‘𝑅) + (𝑃 ↑ 𝑋)) = (𝑃 ↑ 𝑋))
144	79, 141, 143	3eqtrd 2769	. . . 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 2825	. . . . 5 ⊢ (𝜑 → ((𝑃 − 1) + 1) ∈ ℕ₀)
146	41, 13, 43, 5, 7	mulgnn0cld 19054	. . . . 5 ⊢ (𝜑 → (𝑃 ↑ 𝑌) ∈ 𝐵)
147		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → 𝑖 = ((𝑃 − 1) + 1))
148	18	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃 − 1) + 1) = 𝑃)
149	147, 148	eqtrd 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → 𝑖 = 𝑃)
150	149	oveq2d 7433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (𝑃C𝑖) = (𝑃C𝑃))
151	149	oveq2d 7433	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (𝑃 − 𝑖) = (𝑃 − 𝑃))
152	151	oveq1d 7432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃 − 𝑖) ↑ 𝑋) = ((𝑃 − 𝑃) ↑ 𝑋))
153	149	oveq1d 7432	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (𝑖 ↑ 𝑌) = (𝑃 ↑ 𝑌))
154	152, 153	oveq12d 7435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → (((𝑃 − 𝑖) ↑ 𝑋)(._r‘𝑅)(𝑖 ↑ 𝑌)) = (((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌)))
155	150, 154	oveq12d 7435	. . . . . 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 14303	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ₀ → (𝑃C𝑃) = 1)
157	5, 156	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑃C𝑃) = 1)
158	16	subidd 11589	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 − 𝑃) = 0)
159	158	oveq1d 7432	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑃 − 𝑃) ↑ 𝑋) = (0 ↑ 𝑋))
160	41, 127, 13	mulg0 19034	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐵 → (0 ↑ 𝑋) = (1_r‘𝑅))
161	6, 160	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ↑ 𝑋) = (1_r‘𝑅))
162	159, 161	eqtrd 2765	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 − 𝑃) ↑ 𝑋) = (1_r‘𝑅))
163	162	oveq1d 7432	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌)) = ((1_r‘𝑅)(._r‘𝑅)(𝑃 ↑ 𝑌)))
164	8, 9, 126	ringlidm 20209	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝑃 ↑ 𝑌) ∈ 𝐵) → ((1_r‘𝑅)(._r‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
165	39, 146, 164	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((1_r‘𝑅)(._r‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
166	163, 165	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌)) = (𝑃 ↑ 𝑌))
167	157, 166	oveq12d 7435	. . . . . . . 8 ⊢ (𝜑 → ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))) = (1(._g‘𝑅)(𝑃 ↑ 𝑌)))
168	8, 10	mulg1 19040	. . . . . . . . 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 2765	. . . . . . 7 ⊢ (𝜑 → ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))) = (𝑃 ↑ 𝑌))
171	170	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = ((𝑃 − 1) + 1)) → ((𝑃C𝑃)(._g‘𝑅)(((𝑃 − 𝑃) ↑ 𝑋)(._r‘𝑅)(𝑃 ↑ 𝑌))) = (𝑃 ↑ 𝑌))
172	155, 171	eqtrd 2765	. . . . 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 19911	. . . 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 7435	. . 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 2765	. 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 2769	1 ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3945 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6547 (class class class)co 7417 0cc0 11138 1c1 11139 + caddc 11141 − cmin 11474 ℕcn 12242 ℕ₀cn0 12502 ℤcz 12588 ℤ_≥cuz 12852 ...cfz 13516 Ccbc 14293 ∥ cdvds 16230 ℙcprime 16641 Basecbs 17179 +_gcplusg 17232 ._rcmulr 17233 0_gc0g 17420 Σ_g cgsu 17421 Mndcmnd 18693 Grpcgrp 18894 ._gcmg 19027 CMndccmn 19739 mulGrpcmgp 20078 1_rcur 20125 Ringcrg 20177 CRingccrg 20178 chrcchr 21431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-gcd 16469 df-prm 16642 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-cntz 19272 df-od 19487 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-srg 20131 df-ring 20179 df-cring 20180 df-chr 21435

This theorem is referenced by: ply1fermltlchr 22240 frobrhm 32997

W3C validator