gausslemma2dlem4 - Intuitionistic Logic Explorer

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Theorem gausslemma2dlem4 15305

Description: Lemma 4 for gausslemma2d 15310. (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 15302	. 2 ⊢ (𝜑 → (!‘𝐻) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
5		gausslemma2d.m	. . . . . 6 ⊢ 𝑀 = (⌊‘(𝑃 / 4))
6		3lt4 9163	. . . . . . . 8 ⊢ 3 < 4
7		breq1 4036	. . . . . . . 8 ⊢ (𝑃 = 3 → (𝑃 < 4 ↔ 3 < 4))
8	6, 7	mpbiri 168	. . . . . . 7 ⊢ (𝑃 = 3 → 𝑃 < 4)
9		3nn0 9267	. . . . . . . . 9 ⊢ 3 ∈ ℕ₀
10		eleq1 2259	. . . . . . . . 9 ⊢ (𝑃 = 3 → (𝑃 ∈ ℕ₀ ↔ 3 ∈ ℕ₀))
11	9, 10	mpbiri 168	. . . . . . . 8 ⊢ (𝑃 = 3 → 𝑃 ∈ ℕ₀)
12		4nn 9154	. . . . . . . 8 ⊢ 4 ∈ ℕ
13		divfl0 10386	. . . . . . . 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 2241	. . . . 5 ⊢ (𝑃 = 3 → 𝑀 = 0)
17		oveq2 5930	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (1...𝑀) = (1...0))
18	17	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = (1...0))
19		fz10 10121	. . . . . . . . . . 11 ⊢ (1...0) = ∅
20	18, 19	eqtrdi 2245	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝑀) = ∅)
21	20	prodeq1d 11729	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = ∏𝑘 ∈ ∅ (𝑅‘𝑘))
22		prod0 11750	. . . . . . . . 9 ⊢ ∏𝑘 ∈ ∅ (𝑅‘𝑘) = 1
23	21, 22	eqtrdi 2245	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) = 1)
24		oveq1 5929	. . . . . . . . . . . 12 ⊢ (𝑀 = 0 → (𝑀 + 1) = (0 + 1))
25	24	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = (0 + 1))
26		0p1e1 9104	. . . . . . . . . . 11 ⊢ (0 + 1) = 1
27	25, 26	eqtrdi 2245	. . . . . . . . . 10 ⊢ ((𝑀 = 0 ∧ 𝜑) → (𝑀 + 1) = 1)
28	27	oveq1d 5937	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → ((𝑀 + 1)...𝐻) = (1...𝐻))
29	28	prodeq1d 11729	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
30	23, 29	oveq12d 5940	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝜑) → (∏𝑘 ∈ (1...𝑀)(𝑅‘𝑘) · ∏𝑘 ∈ ((𝑀 + 1)...𝐻)(𝑅‘𝑘)) = (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)))
31		1zzd 9353	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℤ)
32	1, 2	gausslemma2dlem0b 15291	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 ∈ ℕ)
33	32	nnzd 9447	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ ℤ)
34	31, 33	fzfigd 10523	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝐻) ∈ Fin)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1...𝐻) ∈ Fin)
36		oveq1 5929	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑘 → (𝑥 · 2) = (𝑘 · 2))
37	36	breq1d 4043	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑘 · 2) < (𝑃 / 2)))
38	36	oveq2d 5938	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑘 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑘 · 2)))
39	37, 36, 38	ifbieq12d 3587	. . . . . . . . . . . . 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 10101	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 𝑘 ∈ ℤ)
42		2z 9354	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
43	42	a1i 9	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → 2 ∈ ℤ)
44	41, 43	zmulcld 9454	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℤ)
45	1	eldifad 3168	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑃 ∈ ℙ)
46		prmz 12279	. . . . . . . . . . . . . . . . 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 9453	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑃 − (𝑘 · 2)) ∈ ℤ)
50		zq 9700	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 · 2) ∈ ℤ → (𝑘 · 2) ∈ ℚ)
51	44, 50	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑘 · 2) ∈ ℚ)
52		2nn 9152	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
53		znq 9698	. . . . . . . . . . . . . . . . 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 10335	. . . . . . . . . . . . . . 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 3597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))) ∈ ℤ)
59	3, 39, 40, 58	fvmptd3 5655	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) = if((𝑘 · 2) < (𝑃 / 2), (𝑘 · 2), (𝑃 − (𝑘 · 2))))
60	59, 58	eqeltrd 2273	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝐻)) → (𝑅‘𝑘) ∈ ℤ)
61	60	zcnd 9449	. . . . . . . . . 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 11772	. . . . . . . 8 ⊢ ((𝑀 = 0 ∧ 𝜑) → ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘) ∈ ℂ)
64	63	mullidd 8044	. . . . . . 7 ⊢ ((𝑀 = 0 ∧ 𝜑) → (1 · ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘)) = ∏𝑘 ∈ (1...𝐻)(𝑅‘𝑘))
65	30, 64	eqtr2d 2230	. . . . . 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 15293	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
70	69	nn0red 9303	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
71	70	ltp1d 8957	. . . . . . 7 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
72		fzdisj 10127	. . . . . . 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 9610	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℤ)
76		znq 9698	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑃 / 4) ∈ ℚ)
77	75, 12, 76	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (𝑃 / 4) ∈ ℚ)
78	77	flqcld 10367	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℤ_≥‘5) → (⌊‘(𝑃 / 4)) ∈ ℤ)
79		nnrp 9738	. . . . . . . . . . . . . . . 16 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
80	12, 79	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ⁺
81		eluzelre 9611	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 𝑃 ∈ ℝ)
82		eluz2 9607	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℤ_≥‘5) ↔ (5 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 5 ≤ 𝑃))
83		4lt5 9166	. . . . . . . . . . . . . . . . . 18 ⊢ 4 < 5
84		4re 9067	. . . . . . . . . . . . . . . . . . 19 ⊢ 4 ∈ ℝ
85		5re 9069	. . . . . . . . . . . . . . . . . . . 20 ⊢ 5 ∈ ℝ
86	85	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 5 ∈ ℝ)
87		zre 9330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℝ)
88	87	adantl 277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((5 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℝ)
89		ltleletr 8108	. . . . . . . . . . . . . . . . . . 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 9798	. . . . . . . . . . . . . . 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 9353	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℤ_≥‘5) → 1 ∈ ℤ)
97		flqge 10372	. . . . . . . . . . . . . . 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 9349	. . . . . . . . . . . . 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 12428	. . . . . . . . . . . 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 3285	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
106		prmuz2 12299	. . . . . . . . . . . . . 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 10396	. . . . . . . . . . . 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 5933	. . . . . . . . 9 ⊢ (1...𝐻) = (1...((𝑃 − 1) / 2))
116	5, 115	eleq12i 2264	. . . . . . . 8 ⊢ (𝑀 ∈ (1...𝐻) ↔ (⌊‘(𝑃 / 4)) ∈ (1...((𝑃 − 1) / 2)))
117		elfz1b 10165	. . . . . . . 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 10126	. . . . . 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 11762	. . . 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 9060	. . . . . . 7 ⊢ 2 ∈ ℝ
127	126	a1i 9	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
128		oddprmgt2 12302	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
129	1, 128	syl 14	. . . . . 6 ⊢ (𝜑 → 2 < 𝑃)
130	127, 129	gtned 8139	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 2)
131	130	neneqd 2388	. . . 4 ⊢ (𝜑 → ¬ 𝑃 = 2)
132		prm23ge5 12433	. . . . . . 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 2229	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 2167 ∖ cdif 3154 ∪ cun 3155 ∩ cin 3156 ∅c0 3450 ifcif 3561 {csn 3622 class class class wbr 4033 ↦ cmpt 4094 ‘cfv 5258 (class class class)co 5922 Fincfn 6799 ℂcc 7877 ℝcr 7878 0cc0 7879 1c1 7880 + caddc 7882 · cmul 7884 < clt 8061 ≤ cle 8062 − cmin 8197 / cdiv 8699 ℕcn 8990 2c2 9041 3c3 9042 4c4 9043 5c5 9044 ℕ₀cn0 9249 ℤcz 9326 ℤ_≥cuz 9601 ℚcq 9693 ℝ⁺crp 9728 ...cfz 10083 ⌊cfl 10358 !cfa 10817 ∏cprod 11715 ℙcprime 12275

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-tp 3630 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-2o 6475 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-5 9052 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-ioo 9967 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-fac 10818 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-proddc 11716 df-dvds 11953 df-prm 12276

This theorem is referenced by: gausslemma2dlem6 15308

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