gausslemma2dlem4 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem4 15108

Description: Lemma 4 for gausslemma2d 15113. (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 15105	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		gausslemma2d.m	. . . . . 6 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
6		3lt4 9140	. . . . . . . 8 ⊢ 3 < 4
7		breq1 4028	. . . . . . . 8 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
8	6, 7	mpbiri 168	. . . . . . 7 ⊢ (𝑃 = 3 → 𝑃 < 4)
9		3nn0 9244	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
10		eleq1 2252	. . . . . . . . 9 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
11	9, 10	mpbiri 168	. . . . . . . 8 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
12		4nn 9131	. . . . . . . 8 ⊢ 4 ∈ ℕ
13		divfl0 10351	. . . . . . . 8 ⊢ ((𝑃 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (𝑃 < 4 ↔ (⌊‘(𝑃 / 4)) = 0))
14	11, 12, 13	sylancl 413	. . . . . . 7 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ (⌊‘(𝑃 / 4)) = 0))
15	8, 14	mpbid 147	. . . . . 6 ⊢ (𝑃 = 3 → (⌊‘(𝑃 / 4)) = 0)
16	5, 15	eqtrid 2234	. . . . 5 ⊢ (𝑃 = 3 → 𝑀 = 0)
17		oveq2 5914	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
18	17	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
19		fz10 10098	. . . . . . . . . . 11 ⊢ (1...0) = ∅
20	18, 19	eqtrdi 2238	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
21	20	prodeq1d 11681	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
22		prod0 11702	. . . . . . . . 9 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
23	21, 22	eqtrdi 2238	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
24		oveq1 5913	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
25	24	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
26		0p1e1 9082	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
27	25, 26	eqtrdi 2238	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
28	27	oveq1d 5921	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
29	28	prodeq1d 11681	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
30	23, 29	oveq12d 5924	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
31		1zzd 9330	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℤ)
32	1, 2	gausslemma2dlem0b 15094	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ)
33	32	nnzd 9424	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℤ)
34	31, 33	fzfigd 10488	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝐻) ∈ Fin)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
36		oveq1 5913	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
37	36	breq1d 4035	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
38	36	oveq2d 5922	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
39	37, 36, 38	ifbieq12d 3579	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
40		simpr 110	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑘 ∈ (1...𝐻))
41	40	elfzelzd 10078	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑘 ∈ ℤ)
42		2z 9331	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
43	42	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 2 ∈ ℤ)
44	41, 43	zmulcld 9431	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℤ)
45	1	eldifad 3160	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℙ)
46		prmz 12223	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
47	45, 46	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
48	47	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑃 ∈ ℤ)
49	48, 44	zsubcld 9430	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℤ)
50		zq 9677	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 · 2) ∈ ℤ → (𝑘 · 2) ∈ ℚ)
51	44, 50	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℚ)
52		2nn 9129	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
53		znq 9675	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑃 / 2) ∈ ℚ)
54	47, 52, 53	sylancl 413	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑃 / 2) ∈ ℚ)
55	54	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 / 2) ∈ ℚ)
56		qdclt 10301	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 · 2) ∈ ℚ ∧ (𝑃 / 2) ∈ ℚ) → DECID (𝑘 · 2) < (𝑃 / 2))
57	51, 55, 56	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → DECID (𝑘 · 2) < (𝑃 / 2))
58	44, 49, 57	ifcldcd 3589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℤ)
59	3, 39, 40, 58	fvmptd3 5639	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2266	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℤ)
61	60	zcnd 9426	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
62	61	adantll 476	. . . . . . . . 9 ⊢ (((𝑀 = 0 ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
63	35, 62	fprodcl 11724	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
64	63	mullidd 8023	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
65	30, 64	eqtr2d 2223	. . . . . 6 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
66	65	ex 115	. . . . 5 ⊢ (𝑀 = 0 → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
67	16, 66	syl 14	. . . 4 ⊢ (𝑃 = 3 → (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘))))
68	67	impcom 125	. . 3 ⊢ ((𝜑 ∧ 𝑃 = 3) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
69	1, 5	gausslemma2dlem0d 15096	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70	69	nn0red 9280	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
71	70	ltp1d 8935	. . . . . . 7 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
72		fzdisj 10104	. . . . . . 7 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
73	71, 72	syl 14	. . . . . 6 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
74	73	adantl 277	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ((1...𝑀) ∩ ((𝑀 + 1)...𝐻)) = ∅)
75		eluzelz 9587	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℤ)
76		znq 9675	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑃 / 4) ∈ ℚ)
77	75, 12, 76	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℚ)
78	77	flqcld 10332	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
79		nnrp 9715	. . . . . . . . . . . . . . . 16 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
80	12, 79	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ⁺
81		eluzelre 9588	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
82		eluz2 9584	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
83		4lt5 9143	. . . . . . . . . . . . . . . . . 18 ⊢ 4 < 5
84		4re 9045	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
85		5re 9047	. . . . . . . . . . . . . . . . . . . 20 ⊢ 5 ∈ ℝ
86	85	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 9307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 8087	. . . . . . . . . . . . . . . . . . 19 ⊢ ((4 ∈ ℝ ∧ 5 ∈ ℝ ∧ 𝑃 ∈ ℝ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
90	84, 86, 88, 89	mp3an2i 1353	. . . . . . . . . . . . . . . . . 18 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
91	83, 90	mpani 430	. . . . . . . . . . . . . . . . 17 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (5 ≤ 𝑃 → 4 ≤ 𝑃))
92	91	3impia 1202	. . . . . . . . . . . . . . . 16 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃)
93	82, 92	sylbi 121	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≤ 𝑃)
94		divge1 9775	. . . . . . . . . . . . . . 15 ⊢ ((4 ∈ ℝ⁺ ∧ 𝑃 ∈ ℝ ∧ 4 ≤ 𝑃) → 1 ≤ (𝑃 / 4))
95	80, 81, 93, 94	mp3an2i 1353	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (𝑃 / 4))
96		1zzd 9330	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flqge 10337	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 / 4) ∈ ℚ ∧ 1 ∈ ℤ) → (1 ≤ (𝑃 / 4) ↔ 1 ≤ (⌊‘(𝑃 / 4))))
98	77, 96, 97	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (1 ≤ (𝑃 / 4) ↔ 1 ≤ (⌊‘(𝑃 / 4))))
99	95, 98	mpbid 147	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (⌊‘(𝑃 / 4)))
100		elnnz1 9326	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑃 / 4)) ∈ ℕ ↔ ((⌊‘(𝑃 / 4)) ∈ ℤ ∧ 1 ≤ (⌊‘(𝑃 / 4))))
101	78, 99, 100	sylanbrc 417	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℕ)
102	101	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → (⌊‘(𝑃 / 4)) ∈ ℕ)
103		oddprm 12371	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
104	103	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → ((𝑃 − 1) / 2) ∈ ℕ)
105		eldifi 3277	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
106		prmuz2 12243	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
107	105, 106	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
108	107	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → 𝑃 ∈ (ℤ_≥‘2))
109		fldiv4lem1div2uz2 10361	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℤ_≥‘2) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
110	108, 109	syl 14	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))
111	102, 104, 110	3jca 1179	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
112	111	ex 115	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∈ (ℤ_≥‘5) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))))
113	1, 112	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ (ℤ_≥‘5) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2))))
114	113	impcom 125	. . . . . . 7 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
115	2	oveq2i 5917	. . . . . . . . 9 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
116	5, 115	eleq12i 2257	. . . . . . . 8 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
117		elfz1b 10142	. . . . . . . 8 ⊢ ((⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)) ↔ ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
118	116, 117	bitri 184	. . . . . . 7 ⊢ (𝑀 ∈ (1...𝐻) ↔ ((⌊‘(𝑃 / 4)) ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ (⌊‘(𝑃 / 4)) ≤ ((𝑃 − 1) / 2)))
119	114, 118	sylibr 134	. . . . . 6 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → 𝑀 ∈ (1...𝐻))
120		fzsplit 10103	. . . . . 6 ⊢ (𝑀 ∈ (1...𝐻) → (1...𝐻) = ((1...𝑀) ∪ ((𝑀 + 1)...𝐻)))
121	119, 120	syl 14	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → (1...𝐻) = ((1...𝑀) ∪ ((𝑀 + 1)...𝐻)))
122	34	adantl 277	. . . . 5 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → (1...𝐻) ∈ Fin)
123	61	adantll 476	. . . . 5 ⊢ (((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℂ)
124	74, 121, 122, 123	fprodsplit 11714	. . . 4 ⊢ ((𝑃 ∈ (ℤ_≥‘5) ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
125	124	ancoms 268	. . 3 ⊢ ((𝜑 ∧ 𝑃 ∈ (ℤ_≥‘5)) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
126		2re 9038	. . . . . . 7 ⊢ 2 ∈ ℝ
127	126	a1i 9	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
128		oddprmgt2 12246	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
129	1, 128	syl 14	. . . . . 6 ⊢ (𝜑 → 2 < 𝑃)
130	127, 129	gtned 8118	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 2)
131	130	neneqd 2381	. . . 4 ⊢ (𝜑 → ¬ 𝑃 = 2)
132		prm23ge5 12376	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
133	45, 132	syl 14	. . . . . 6 ⊢ (𝜑 → (𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
134		3orass 983	. . . . . 6 ⊢ ((𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)) ↔ (𝑃 = 2 ∨ (𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5))))
135	133, 134	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑃 = 2 ∨ (𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5))))
136	135	ord 725	. . . 4 ⊢ (𝜑 → (¬ 𝑃 = 2 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5))))
137	131, 136	mpd 13	. . 3 ⊢ (𝜑 → (𝑃 = 3 ∨ 𝑃 ∈ (ℤ_≥‘5)))
138	68, 125, 137	mpjaodan 799	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
139	4, 138	eqtrd 2222	1 ⊢ (𝜑 → (!‘𝐻) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∨ w3o 979 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∖ cdif 3146 ∪ cun 3147 ∩ cin 3148 ∅c0 3442 ifcif 3553 {csn 3614 class class class wbr 4025 ↦ cmpt 4086 ‘cfv 5242 (class class class)co 5906 Fincfn 6781 ℂcc 7856 ℝcr 7857 0cc0 7858 1c1 7859 + caddc 7861 · cmul 7863 < clt 8040 ≤ cle 8041 − cmin 8176 / cdiv 8677 ℕcn 8968 2c2 9019 3c3 9020 4c4 9021 5c5 9022 ℕ₀cn0 9226 ℤcz 9303 ℤ_≥cuz 9578 ℚcq 9670 ℝ⁺crp 9705 ...cfz 10060 ⌊cfl 10323 !cfa 10770 ∏cprod 11667 ℙcprime 12219

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 ax-caucvg 7978

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-tp 3622 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-ilim 4394 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-isom 5251 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-recs 6345 df-irdg 6410 df-frec 6431 df-1o 6456 df-2o 6457 df-oadd 6460 df-er 6574 df-en 6782 df-dom 6783 df-fin 6784 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-2 9027 df-3 9028 df-4 9029 df-5 9030 df-n0 9227 df-z 9304 df-uz 9579 df-q 9671 df-rp 9706 df-ioo 9944 df-fz 10061 df-fzo 10195 df-fl 10325 df-mod 10380 df-seqfrec 10505 df-exp 10584 df-fac 10771 df-ihash 10821 df-cj 10960 df-re 10961 df-im 10962 df-rsqrt 11116 df-abs 11117 df-clim 11396 df-proddc 11668 df-dvds 11905 df-prm 12220

This theorem is referenced by: gausslemma2dlem6 15111

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