gausslemma2dlem4 - Metamath Proof Explorer

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Theorem gausslemma2dlem4 27346

Description: Lemma 4 for gausslemma2d 27351. (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 27343	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		eldif 3900	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∈ {2}))
6		prm23ge5 16777	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
7		eleq1 2825	. . . . . . . . 9 ⊢ (𝑃 = 2 → (𝑃 ∈ {2} ↔ 2 ∈ {2}))
8	7	notbid 318	. . . . . . . 8 ⊢ (𝑃 = 2 → (¬ 𝑃 ∈ {2} ↔ ¬ 2 ∈ {2}))
9		2ex 12249	. . . . . . . . . . . 12 ⊢ 2 ∈ V
10	9	snid 4607	. . . . . . . . . . 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 2951	. . . . . . . . 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 240	. . . . . . 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 12341	. . . . . . . . . . . 12 ⊢ 3 < 4
17		breq1 5089	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
18	16, 17	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑃 = 3 → 𝑃 < 4)
19		3nn0 12446	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
21	19, 20	mpbiri 258	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
22		4nn 12255	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
23		divfl0 13774	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (𝑃 < 4 ↔ (⌊‘(𝑃 / 4)) = 0))
24	21, 22, 23	sylancl 587	. . . . . . . . . . 11 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ (⌊‘(𝑃 / 4)) = 0))
25	18, 24	mpbid 232	. . . . . . . . . 10 ⊢ (𝑃 = 3 → (⌊‘(𝑃 / 4)) = 0)
26	15, 25	eqtrid 2784	. . . . . . . . 9 ⊢ (𝑃 = 3 → 𝑀 = 0)
27		oveq2 7368	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
28	27	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
29		fz10 13490	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
30	28, 29	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
31	30	prodeq1d 15876	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
32		prod0 15899	. . . . . . . . . . . . 13 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
33	31, 32	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
34		oveq1 7367	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
36		0p1e1 12289	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
37	35, 36	eqtrdi 2788	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
38	37	oveq1d 7375	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
39	38	prodeq1d 15876	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
40	33, 39	oveq12d 7378	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
41		fzfid 13926	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
42		oveq1 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
43	42	breq1d 5096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
44	42	oveq2d 7376	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
45	43, 42, 44	ifbieq12d 4496	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑘 ∈ (1...𝐻))
47		elfzelz 13469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℤ)
48	47	zcnd 12625	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℂ)
49		2cnd 12250	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 2 ∈ ℂ)
50	48, 49	mulcld 11156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐻) → (𝑘 · 2) ∈ ℂ)
51	50	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℂ)
52		eldifi 4072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
53		prmz 16635	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
54	53	zcnd 12625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
55	1, 52, 54	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ ℂ)
56	55	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑃 ∈ ℂ)
57	56, 51	subcld 11496	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℂ)
58	51, 57	ifcld 4514	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℂ)
59	3, 45, 46, 58	fvmptd3 6965	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
61	60	adantll 715	. . . . . . . . . . . . 13 ⊢ (((𝑀 = 0 ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
62	41, 61	fprodcl 15908	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
63	62	mullidd 11154	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
64	40, 63	eqtr2d 2773	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
65	64	ex 412	. . . . . . . . 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 27336	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
69	68	nn0red 12490	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
70	69	ltp1d 12077	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
71		fzdisj 13496	. . . . . . . . . . . 12 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
72	70, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
73	72	adantl 481	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
74		eluzelre 12790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
75		4re 12256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ∈ ℝ
76	75	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ∈ ℝ)
77		4ne0 12280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ≠ 0
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≠ 0)
79	74, 76, 78	redivcld 11974	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℝ)
80	79	flcld 13748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
81		nnrp 12945	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
82	22, 81	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 4 ∈ ℝ⁺
83		eluz2 12785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
84		4lt5 12344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 4 < 5
85		5re 12259	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 5 ∈ ℝ
86	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 12519	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 11230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 ∈ ℝ ∧ 5 ∈ ℝ ∧ 𝑃 ∈ ℝ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
90	75, 86, 88, 89	mp3an2i 1469	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
91	84, 90	mpani 697	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (5 ≤ 𝑃 → 4 ≤ 𝑃))
92	91	3impia 1118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃)
93	83, 92	sylbi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≤ 𝑃)
94		divge1 13003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((4 ∈ ℝ⁺ ∧ 𝑃 ∈ ℝ ∧ 4 ≤ 𝑃) → 1 ≤ (𝑃 / 4))
95	82, 74, 93, 94	mp3an2i 1469	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (𝑃 / 4))
96		1zzd 12549	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flge 13755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 / 4) ∈ ℝ ∧ 1 ∈ ℤ) → (1 ≤ (𝑃 / 4) ↔ 1 ≤ (⌊‘(𝑃 / 4))))
98	79, 96, 97	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (1 ≤ (𝑃 / 4) ↔ 1 ≤ (⌊‘(𝑃 / 4))))
99	95, 98	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (⌊‘(𝑃 / 4)))
100		elnnz1 12544	. . . . . . . . . . . . . . . . . 18 ⊢ ((⌊‘(𝑃 / 4)) ∈ ℕ ↔ ((⌊‘(𝑃 / 4)) ∈ ℤ ∧ 1 ≤ (⌊‘(𝑃 / 4))))
101	80, 99, 100	sylanbrc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℕ)
102	101	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → (⌊‘(𝑃 / 4)) ∈ ℕ)
103		oddprm 16772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
104	103	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → ((𝑃 − 1) / 2) ∈ ℕ)
105		prmuz2 16656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
106	52, 105	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
107	106	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → 𝑃 ∈ (ℤ_≥‘2))
108		fldiv4lem1div2uz2 13786	. . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
111	110	ex 412	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
114	2	oveq2i 7371	. . . . . . . . . . . . . 14 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
115	15, 114	eleq12i 2830	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
116		elfz1b 13538	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)) ↔ ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
117	115, 116	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (1...𝐻) ↔ ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
118	113, 117	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → 𝑀 ∈ (1...𝐻))
119		fzsplit 13495	. . . . . . . . . . 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 13926	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → (1...𝐻) ∈ Fin)
122	60	adantll 715	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
123	73, 120, 121, 122	fprodsplit 15922	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
124	123	ex 412	. . . . . . . 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 1431	. . . . . 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 406	. . . 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 217	. . 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 2772	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 206 ∧ wa 395 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ∅c0 4274 ifcif 4467 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 2c2 12227 3c3 12228 4c4 12229 5c5 12230 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ℝ⁺crp 12933 ...cfz 13452 ⌊cfl 13740 !cfa 14226 ∏cprod 15859 ℙcprime 16631

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-ioo 13293 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-fac 14227 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-prod 15860 df-dvds 16213 df-prm 16632

This theorem is referenced by: gausslemma2dlem6 27349

W3C validator