gausslemma2dlem4 - Metamath Proof Explorer

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Theorem gausslemma2dlem4 27413

Description: Lemma 4 for gausslemma2d 27418. (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 27410	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		eldif 3961	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∈ {2}))
6		prm23ge5 16853	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
7		eleq1 2829	. . . . . . . . 9 ⊢ (𝑃 = 2 → (𝑃 ∈ {2} ↔ 2 ∈ {2}))
8	7	notbid 318	. . . . . . . 8 ⊢ (𝑃 = 2 → (¬ 𝑃 ∈ {2} ↔ ¬ 2 ∈ {2}))
9		2ex 12343	. . . . . . . . . . . 12 ⊢ 2 ∈ V
10	9	snid 4662	. . . . . . . . . . 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 2958	. . . . . . . . 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 12440	. . . . . . . . . . . 12 ⊢ 3 < 4
17		breq1 5146	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
18	16, 17	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑃 = 3 → 𝑃 < 4)
19		3nn0 12544	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20		eleq1 2829	. . . . . . . . . . . . 13 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
21	19, 20	mpbiri 258	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
22		4nn 12349	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
23		divfl0 13864	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝑃 = 3 → (⌊‘(𝑃 / 4)) = 0)
26	15, 25	eqtrid 2789	. . . . . . . . 9 ⊢ (𝑃 = 3 → 𝑀 = 0)
27		oveq2 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
28	27	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
29		fz10 13585	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
30	28, 29	eqtrdi 2793	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
31	30	prodeq1d 15956	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
32		prod0 15979	. . . . . . . . . . . . 13 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
33	31, 32	eqtrdi 2793	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
34		oveq1 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
36		0p1e1 12388	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
37	35, 36	eqtrdi 2793	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
38	37	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
39	38	prodeq1d 15956	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
40	33, 39	oveq12d 7449	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
41		fzfid 14014	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
42		oveq1 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
43	42	breq1d 5153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
44	42	oveq2d 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
45	43, 42, 44	ifbieq12d 4554	. . . . . . . . . . . . . . . 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 13564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℤ)
48	47	zcnd 12723	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℂ)
49		2cnd 12344	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 2 ∈ ℂ)
50	48, 49	mulcld 11281	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐻) → (𝑘 · 2) ∈ ℂ)
51	50	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℂ)
52		eldifi 4131	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
53		prmz 16712	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
54	53	zcnd 12723	. . . . . . . . . . . . . . . . . . . 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 11620	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℂ)
58	51, 57	ifcld 4572	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℂ)
59	3, 45, 46, 58	fvmptd3 7039	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
61	60	adantll 714	. . . . . . . . . . . . 13 ⊢ (((𝑀 = 0 ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
62	41, 61	fprodcl 15988	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
63	62	mullidd 11279	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
64	40, 63	eqtr2d 2778	. . . . . . . . . 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 27403	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
69	68	nn0red 12588	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
70	69	ltp1d 12198	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
71		fzdisj 13591	. . . . . . . . . . . 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 12889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
75		4re 12350	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ∈ ℝ
76	75	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ∈ ℝ)
77		4ne0 12374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ≠ 0
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≠ 0)
79	74, 76, 78	redivcld 12095	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℝ)
80	79	flcld 13838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
81		nnrp 13046	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
82	22, 81	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 4 ∈ ℝ⁺
83		eluz2 12884	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
84		4lt5 12443	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 4 < 5
85		5re 12353	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 5 ∈ ℝ
86	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 12617	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 11354	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((4 ∈ ℝ ∧ 5 ∈ ℝ ∧ 𝑃 ∈ ℝ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
90	75, 86, 88, 89	mp3an2i 1468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
91	84, 90	mpani 696	. . . . . . . . . . . . . . . . . . . . . 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 13103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((4 ∈ ℝ⁺ ∧ 𝑃 ∈ ℝ ∧ 4 ≤ 𝑃) → 1 ≤ (𝑃 / 4))
95	82, 74, 93, 94	mp3an2i 1468	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (𝑃 / 4))
96		1zzd 12648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flge 13845	. . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (⌊‘(𝑃 / 4)))
100		elnnz1 12643	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → (⌊‘(𝑃 / 4)) ∈ ℕ)
103		oddprm 16848	. . . . . . . . . . . . . . . . 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 16733	. . . . . . . . . . . . . . . . . . 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 13876	. . . . . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . 14 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
115	15, 114	eleq12i 2834	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
116		elfz1b 13633	. . . . . . . . . . . . 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 13590	. . . . . . . . . . 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 14014	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → (1...𝐻) ∈ Fin)
122	60	adantll 714	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
123	73, 120, 121, 122	fprodsplit 16002	. . . . . . . . 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 1430	. . . . . 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 2777	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 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 ifcif 4525 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 4c4 12323 5c5 12324 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ℝ⁺crp 13034 ...cfz 13547 ⌊cfl 13830 !cfa 14312 ∏cprod 15939 ℙcprime 16708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ioo 13391 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-dvds 16291 df-prm 16709

This theorem is referenced by: gausslemma2dlem6 27416

W3C validator