gausslemma2dlem4 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem4 15786

Description: Lemma 4 for gausslemma2d 15791. (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 15783	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		gausslemma2d.m	. . . . . 6 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
6		3lt4 9309	. . . . . . . 8 ⊢ 3 < 4
7		breq1 4089	. . . . . . . 8 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
8	6, 7	mpbiri 168	. . . . . . 7 ⊢ (𝑃 = 3 → 𝑃 < 4)
9		3nn0 9413	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
10		eleq1 2292	. . . . . . . . 9 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
11	9, 10	mpbiri 168	. . . . . . . 8 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
12		4nn 9300	. . . . . . . 8 ⊢ 4 ∈ ℕ
13		divfl0 10549	. . . . . . . 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 2274	. . . . 5 ⊢ (𝑃 = 3 → 𝑀 = 0)
17		oveq2 6021	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
18	17	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
19		fz10 10274	. . . . . . . . . . 11 ⊢ (1...0) = ∅
20	18, 19	eqtrdi 2278	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
21	20	prodeq1d 12118	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
22		prod0 12139	. . . . . . . . 9 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
23	21, 22	eqtrdi 2278	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
24		oveq1 6020	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
25	24	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
26		0p1e1 9250	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
27	25, 26	eqtrdi 2278	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
28	27	oveq1d 6028	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
29	28	prodeq1d 12118	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
30	23, 29	oveq12d 6031	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
31		1zzd 9499	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℤ)
32	1, 2	gausslemma2dlem0b 15772	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ)
33	32	nnzd 9594	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℤ)
34	31, 33	fzfigd 10686	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝐻) ∈ Fin)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
36		oveq1 6020	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
37	36	breq1d 4096	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
38	36	oveq2d 6029	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
39	37, 36, 38	ifbieq12d 3630	. . . . . . . . . . . . 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 10254	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑘 ∈ ℤ)
42		2z 9500	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
43	42	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 2 ∈ ℤ)
44	41, 43	zmulcld 9601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℤ)
45	1	eldifad 3209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℙ)
46		prmz 12676	. . . . . . . . . . . . . . . . 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 9600	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℤ)
50		zq 9853	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 · 2) ∈ ℤ → (𝑘 · 2) ∈ ℚ)
51	44, 50	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℚ)
52		2nn 9298	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
53		znq 9851	. . . . . . . . . . . . . . . . 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 10498	. . . . . . . . . . . . . . 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 3641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℤ)
59	3, 39, 40, 58	fvmptd3 5736	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2306	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℤ)
61	60	zcnd 9596	. . . . . . . . . 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 12161	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
64	63	mullidd 8190	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
65	30, 64	eqtr2d 2263	. . . . . 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 15774	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70	69	nn0red 9449	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
71	70	ltp1d 9103	. . . . . . 7 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
72		fzdisj 10280	. . . . . . 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 9758	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℤ)
76		znq 9851	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑃 / 4) ∈ ℚ)
77	75, 12, 76	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℚ)
78	77	flqcld 10530	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
79		nnrp 9891	. . . . . . . . . . . . . . . 16 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
80	12, 79	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ⁺
81		eluzelre 9759	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
82		eluz2 9754	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
83		4lt5 9312	. . . . . . . . . . . . . . . . . 18 ⊢ 4 < 5
84		4re 9213	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
85		5re 9215	. . . . . . . . . . . . . . . . . . . 20 ⊢ 5 ∈ ℝ
86	85	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 9476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 8254	. . . . . . . . . . . . . . . . . . 19 ⊢ ((4 ∈ ℝ ∧ 5 ∈ ℝ ∧ 𝑃 ∈ ℝ) → ((4 < 5 ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃))
90	84, 86, 88, 89	mp3an2i 1376	. . . . . . . . . . . . . . . . . 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 1224	. . . . . . . . . . . . . . . 16 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃) → 4 ≤ 𝑃)
93	82, 92	sylbi 121	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 4 ≤ 𝑃)
94		divge1 9951	. . . . . . . . . . . . . . 15 ⊢ ((4 ∈ ℝ⁺ ∧ 𝑃 ∈ ℝ ∧ 4 ≤ 𝑃) → 1 ≤ (𝑃 / 4))
95	80, 81, 93, 94	mp3an2i 1376	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ≤ (𝑃 / 4))
96		1zzd 9499	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flqge 10535	. . . . . . . . . . . . . . 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 9495	. . . . . . . . . . . . 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 12825	. . . . . . . . . . . 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 3327	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
106		prmuz2 12696	. . . . . . . . . . . . . 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 10559	. . . . . . . . . . . 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 1201	. . . . . . . . . 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 6024	. . . . . . . . 9 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
116	5, 115	eleq12i 2297	. . . . . . . 8 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
117		elfz1b 10318	. . . . . . . 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 10279	. . . . . 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 12151	. . . 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 9206	. . . . . . 7 ⊢ 2 ∈ ℝ
127	126	a1i 9	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
128		oddprmgt2 12699	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
129	1, 128	syl 14	. . . . . 6 ⊢ (𝜑 → 2 < 𝑃)
130	127, 129	gtned 8285	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 2)
131	130	neneqd 2421	. . . 4 ⊢ (𝜑 → ¬ 𝑃 = 2)
132		prm23ge5 12830	. . . . . . 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 1005	. . . . . 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 729	. . . 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 803	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) = (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)))
139	4, 138	eqtrd 2262	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 713 DECID wdc 839 ∨ w3o 1001 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ∖ cdif 3195 ∪ cun 3196 ∩ cin 3197 ∅c0 3492 ifcif 3603 {csn 3667 class class class wbr 4086 ↦ cmpt 4148 ‘cfv 5324 (class class class)co 6013 Fincfn 6904 ℂcc 8023 ℝcr 8024 0cc0 8025 1c1 8026 + caddc 8028 · cmul 8030 < clt 8207 ≤ cle 8208 − cmin 8343 / cdiv 8845 ℕcn 9136 2c2 9187 3c3 9188 4c4 9189 5c5 9190 ℕ₀cn0 9395 ℤcz 9472 ℤ_≥cuz 9748 ℚcq 9846 ℝ⁺crp 9881 ...cfz 10236 ⌊cfl 10521 !cfa 10980 ∏cprod 12104 ℙcprime 12672

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-tp 3675 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-ioo 10120 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-fac 10981 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-proddc 12105 df-dvds 12342 df-prm 12673

This theorem is referenced by: gausslemma2dlem6 15789

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