gausslemma2dlem4 - Metamath Proof Explorer

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Theorem gausslemma2dlem4 27280

Description: Lemma 4 for gausslemma2d 27285. (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 27277	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		eldif 3924	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 𝑃 ∈ {2}))
6		prm23ge5 16786	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
7		eleq1 2816	. . . . . . . . 9 ⊢ (𝑃 = 2 → (𝑃 ∈ {2} ↔ 2 ∈ {2}))
8	7	notbid 318	. . . . . . . 8 ⊢ (𝑃 = 2 → (¬ 𝑃 ∈ {2} ↔ ¬ 2 ∈ {2}))
9		2ex 12263	. . . . . . . . . . . 12 ⊢ 2 ∈ V
10	9	snid 4626	. . . . . . . . . . 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 2943	. . . . . . . . 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 12355	. . . . . . . . . . . 12 ⊢ 3 < 4
17		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
18	16, 17	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑃 = 3 → 𝑃 < 4)
19		3nn0 12460	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℕ₀
20		eleq1 2816	. . . . . . . . . . . . 13 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
21	19, 20	mpbiri 258	. . . . . . . . . . . 12 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
22		4nn 12269	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
23		divfl0 13786	. . . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (𝑃 = 3 → 𝑀 = 0)
27		oveq2 7395	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
28	27	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
29		fz10 13506	. . . . . . . . . . . . . . 15 ⊢ (1...0) = ∅
30	28, 29	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
31	30	prodeq1d 15886	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
32		prod0 15909	. . . . . . . . . . . . 13 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
33	31, 32	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
34		oveq1 7394	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
36		0p1e1 12303	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
37	35, 36	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
38	37	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
39	38	prodeq1d 15886	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
40	33, 39	oveq12d 7405	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
41		fzfid 13938	. . . . . . . . . . . . 13 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
42		oveq1 7394	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
43	42	breq1d 5117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
44	42	oveq2d 7403	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
45	43, 42, 44	ifbieq12d 4517	. . . . . . . . . . . . . . . 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 13485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℤ)
48	47	zcnd 12639	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 𝑘 ∈ ℂ)
49		2cnd 12264	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (1...𝐻) → 2 ∈ ℂ)
50	48, 49	mulcld 11194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝐻) → (𝑘 · 2) ∈ ℂ)
51	50	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℂ)
52		eldifi 4094	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
53		prmz 16645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
54	53	zcnd 12639	. . . . . . . . . . . . . . . . . . . 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 11533	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℂ)
58	51, 57	ifcld 4535	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℂ)
59	3, 45, 46, 58	fvmptd3 6991	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2828	. . . . . . . . . . . . . 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 15918	. . . . . . . . . . . 12 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
63	62	mullidd 11192	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
64	40, 63	eqtr2d 2765	. . . . . . . . . 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 27270	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
69	68	nn0red 12504	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
70	69	ltp1d 12113	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
71		fzdisj 13512	. . . . . . . . . . . 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 12804	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
75		4re 12270	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ∈ ℝ
76	75	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ∈ ℝ)
77		4ne0 12294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 4 ≠ 0
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≠ 0)
79	74, 76, 78	redivcld 12010	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℝ)
80	79	flcld 13760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
81		nnrp 12963	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
82	22, 81	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 4 ∈ ℝ⁺
83		eluz2 12799	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
84		4lt5 12358	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 4 < 5
85		5re 12273	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 5 ∈ ℝ
86	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 12533	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 11267	. . . . . . . . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃)
93	83, 92	sylbi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≤ 𝑃)
94		divge1 13021	. . . . . . . . . . . . . . . . . . . 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 12564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flge 13767	. . . . . . . . . . . . . . . . . . . 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 12559	. . . . . . . . . . . . . . . . . 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 16781	. . . . . . . . . . . . . . . . 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 16666	. . . . . . . . . . . . . . . . . . 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 13798	. . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 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 7398	. . . . . . . . . . . . . 14 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
115	15, 114	eleq12i 2821	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
116		elfz1b 13554	. . . . . . . . . . . . 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 13511	. . . . . . . . . . 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 13938	. . . . . . . . . 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 15932	. . . . . . . . 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 2764	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 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ∅c0 4296 ifcif 4488 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 3c3 12242 4c4 12243 5c5 12244 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 ⌊cfl 13752 !cfa 14238 ∏cprod 15869 ℙcprime 16641

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-ioo 13310 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-fac 14239 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-prod 15870 df-dvds 16223 df-prm 16642

This theorem is referenced by: gausslemma2dlem6 27283

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