gausslemma2dlem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > gausslemma2dlem4		Structured version Visualization version GIF version

Theorem gausslemma2dlem4 26517

Description: Lemma 4 for gausslemma2d 26522. (Contributed by AV, 16-Jun-2021.)

**Hypotheses**
Ref	Expression
gausslemma2d.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2d.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2d.r	⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
gausslemma2d.m	⊢ 𝑀 = (⌊‘(𝑃 / 4))

**Assertion**
Ref	Expression
gausslemma2dlem4	⊢ (𝜑 → (!‘𝐻) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 𝑘,𝐻 𝑅,𝑘 𝜑,𝑘 𝑥,𝑀,𝑘 𝑃,𝑘 Allowed substitution hint: 𝑅(𝑥)

**Proof of Theorem gausslemma2dlem4**
Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2		gausslemma2d.h	. . 3 ⊢ 𝐻 = ((𝑃 − 1) / 2)
3		gausslemma2d.r	. . 3 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
4	1, 2, 3	gausslemma2dlem1 26514	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		eldif 3897	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∈ {2}))
6		prm23ge5 16516	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
7		eleq1 2826	. . . . . . . . 9 ⊢ (𝑃 = 2 → (𝑃 ∈ {2} ↔ 2 ∈ {2}))
8	7	notbid 318	. . . . . . . 8 ⊢ (𝑃 = 2 → (¬ 𝑃 ∈ {2} ↔ ¬ 2 ∈ {2}))
9		2ex 12050	. . . . . . . . . . . 12 ⊢ 2 ∈ V
10	9	snid 4597	. . . . . . . . . . 11 ⊢ 2 ∈ {2}
11	10	2a1i 12	. . . . . . . . . 10 ⊢ (𝑃 = 2 → (∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ≠ (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) → 2 ∈ {2}))
12	11	necon1bd 2961	. . . . . . . . 9 ⊢ (𝑃 = 2 → (¬ 2 ∈ {2} → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
13	12	a1dd 50	. . . . . . . 8 ⊢ (𝑃 = 2 → (¬ 2 ∈ {2} → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))))
14	8, 13	sylbid 239	. . . . . . 7 ⊢ (𝑃 = 2 → (¬ 𝑃 ∈ {2} → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))))
15		gausslemma2d.m	. . . . . . . . . 10 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
16		3lt4 12147	. . . . . . . . . . . 12 ⊢ 3 < 4
17		breq1 5077	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
18	16, 17	mpbiri 257	. . . . . . . . . . 11 ⊢ (𝑃 = 3 → 𝑃 < 4)
19		3nn0 12251	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
21	19, 20	mpbiri 257	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
22		4nn 12056	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
23		divfl0 13544	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (𝑃 < 4 ↔ (⌊‘(𝑃 / 4)) = 0))
24	21, 22, 23	sylancl 586	. . . . . . . . . . 11 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ (⌊‘(𝑃 / 4)) = 0))
25	18, 24	mpbid 231	. . . . . . . . . 10 ⊢ (𝑃 = 3 → (⌊‘(𝑃 / 4)) = 0)
26	15, 25	eqtrid 2790	. . . . . . . . 9 ⊢ (𝑃 = 3 → 𝑀 = 0)
27		oveq2 7283	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
28	27	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
29		fz10 13277	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
30	28, 29	eqtrdi 2794	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
31	30	prodeq1d 15631	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
32		prod0 15653	. . . . . . . . . . . . 13 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
33	31, 32	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
34		oveq1 7282	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
35	34	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
36		0p1e1 12095	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
37	35, 36	eqtrdi 2794	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
38	37	oveq1d 7290	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
39	38	prodeq1d 15631	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
40	33, 39	oveq12d 7293	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
41		fzfid 13693	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
42		oveq1 7282	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
43	42	breq1d 5084	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
44	42	oveq2d 7291	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
45	43, 42, 44	ifbieq12d 4487	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑘 → if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
46		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑘 ∈ (1...𝐻))
47		elfzelz 13256	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℤ)
48	47	zcnd 12427	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℂ)
49		2cnd 12051	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 2 ∈ ℂ)
50	48, 49	mulcld 10995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐻) → (𝑘 · 2) ∈ ℂ)
51	50	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℂ)
52		eldifi 4061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
53		prmz 16380	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
54	53	zcnd 12427	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
55	1, 52, 54	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℂ)
56	55	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑃 ∈ ℂ)
57	56, 51	subcld 11332	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℂ)
58	51, 57	ifcld 4505	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℂ)
59	3, 45, 46, 58	fvmptd3 6898	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
61	60	adantll 711	. . . . . . . . . . . . 13 ⊢ (((𝑀 = 0 ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
62	41, 61	fprodcl 15662	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
63	62	mulid2d 10993	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
64	40, 63	eqtr2d 2779	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
65	64	ex 413	. . . . . . . . 9 ⊢ (𝑀 = 0 → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
66	26, 65	syl 17	. . . . . . . 8 ⊢ (𝑃 = 3 → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
67	66	a1d 25	. . . . . . 7 ⊢ (𝑃 = 3 → (¬ 𝑃 ∈ {2} → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))))
68	1, 15	gausslemma2dlem0d 26507	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
69	68	nn0red 12294	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
70	69	ltp1d 11905	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
71		fzdisj 13283	. . . . . . . . . . . 12 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
72	70, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
73	72	adantl 482	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
74		eluzelre 12593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
75		4re 12057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ∈ ℝ
76	75	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ∈ ℝ)
77		4ne0 12081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ≠ 0
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≠ 0)
79	74, 76, 78	redivcld 11803	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℝ)
80	79	flcld 13518	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
81		nnrp 12741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
82	22, 81	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 4 ∈ ℝ⁺
83		eluz2 12588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
84		4lt5 12150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 4 < 5
85		5re 12060	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 5 ∈ ℝ
86	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 12323	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 11068	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 ∈ ℝ ∧ 5 ∈ ℝ ∧ 𝑃 ∈ ℝ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
90	75, 86, 88, 89	mp3an2i 1465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
91	84, 90	mpani 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (5 ≤ 𝑃 → 4 ≤ 𝑃))
92	91	3impia 1116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃)
93	83, 92	sylbi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≤ 𝑃)
94		divge1 12798	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((4 ∈ ℝ⁺ ∧ 𝑃 ∈ ℝ ∧ 4 ≤ 𝑃) → 1 ≤ (𝑃 / 4))
95	82, 74, 93, 94	mp3an2i 1465	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (𝑃 / 4))
96		1zzd 12351	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flge 13525	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 / 4) ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ (𝑃 / 4) ↔ 1 ≤ (⌊‘(𝑃 / 4))))
98	79, 96, 97	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (1 ≤ (𝑃 / 4) ↔ 1 ≤ (⌊‘(𝑃 / 4))))
99	95, 98	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (⌊‘(𝑃 / 4)))
100		elnnz1 12346	. . . . . . . . . . . . . . . . . 18 ⊢ ((⌊‘(𝑃 / 4)) ∈ ℕ ↔ ((⌊‘(𝑃 / 4)) ∈ ℤ ∧ 1 ≤ (⌊‘(𝑃 / 4))))
101	80, 99, 100	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℕ)
102	101	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → (⌊‘(𝑃 / 4)) ∈ ℕ)
103		oddprm 16511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
104	103	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → ((𝑃 − 1) / 2) ∈ ℕ)
105		prmuz2 16401	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
106	52, 105	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
107	106	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → 𝑃 ∈ (ℤ_≥‘2))
108		fldiv4lem1div2uz2 13556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
109	107, 108	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
110	102, 104, 109	3jca 1127	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
111	110	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∈ (ℤ_≥‘5) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))))
112	1, 111	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘5) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))))
113	112	impcom 408	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
114	2	oveq2i 7286	. . . . . . . . . . . . . 14 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
115	15, 114	eleq12i 2831	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
116		elfz1b 13325	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)) ↔ ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
117	115, 116	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (1...𝐻) ↔ ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
118	113, 117	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → 𝑀 ∈ (1...𝐻))
119		fzsplit 13282	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (1...𝐻) → (1...𝐻) = ((1...𝑀) ∪ ((𝑀 + 1)...𝐻)))
120	118, 119	syl 17	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → (1...𝐻) = ((1...𝑀) ∪ ((𝑀 + 1)...𝐻)))
121		fzfid 13693	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → (1...𝐻) ∈ Fin)
122	60	adantll 711	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
123	73, 120, 121, 122	fprodsplit 15676	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
124	123	ex 413	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
125	124	a1d 25	. . . . . . 7 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (¬ 𝑃 ∈ {2} → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))))
126	14, 67, 125	3jaoi 1426	. . . . . 6 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)) → (¬ 𝑃 ∈ {2} → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))))
127	6, 126	syl 17	. . . . 5 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 ∈ {2} → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))))
128	127	imp 407	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 ∈ {2}) → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
129	5, 128	sylbi 216	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
130	1, 129	mpcom 38	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
131	4, 130	eqtrd 2778	1 ⊢ (𝜑 → (!‘𝐻) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ w3o 1085 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ∅c0 4256 ifcif 4459 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 3c3 12029 4c4 12030 5c5 12031 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ...cfz 13239 ⌊cfl 13510 !cfa 13987 ∏cprod 15615 ℙcprime 16376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ioo 13083 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-fac 13988 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-prod 15616 df-dvds 15964 df-prm 16377

This theorem is referenced by: gausslemma2dlem6 26520

W3C validator