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Theorem pilem2 25811

Description: Lemma for pire 25815, pigt2lt4 25813 and sinpi 25814. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)

**Hypotheses**
Ref	Expression
pilem2.1	⊢ (𝜑 → 𝐴 ∈ (2(,)4))
pilem2.2	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pilem2.3	⊢ (𝜑 → (sin‘𝐴) = 0)
pilem2.4	⊢ (𝜑 → (sin‘𝐵) = 0)

**Assertion**
Ref	Expression
pilem2	⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)

Proof of Theorem pilem2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pi 15955	. . . 4 ⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
2		inss1 4188	. . . . . . 7 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ⁺
3		rpssre 12922	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
4	2, 3	sstri 3953	. . . . . 6 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ)
6		0re 11157	. . . . . . 7 ⊢ 0 ∈ ℝ
7		elinel1 4155	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 𝑦 ∈ ℝ⁺)
8	7	rpge0d 12961	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 0 ≤ 𝑦)
9	8	rgen 3066	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦
10		breq1 5108	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
11	10	ralbidv 3174	. . . . . . . 8 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦))
12	11	rspcev 3581	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦)
13	6, 9, 12	mp2an 690	. . . . . 6 ⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦)
15		2re 12227	. . . . . . . . 9 ⊢ 2 ∈ ℝ
16		pilem2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
17	16	rpred 12957	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
18		remulcl 11136	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 · 𝐵) ∈ ℝ)
19	15, 17, 18	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
20		pilem2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (2(,)4))
21		elioore 13294	. . . . . . . . 9 ⊢ (𝐴 ∈ (2(,)4) → 𝐴 ∈ ℝ)
22	20, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
23	19, 22	resubcld 11583	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ)
24		4re 12237	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ∈ ℝ)
26		eliooord 13323	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (2(,)4) → (2 < 𝐴 ∧ 𝐴 < 4))
27	20, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝐴 ∧ 𝐴 < 4))
28	27	simprd 496	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 4)
29		2t2e4 12317	. . . . . . . . . 10 ⊢ (2 · 2) = 4
30	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ)
31		0red 11158	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ∈ ℝ)
32		2pos 12256	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
33	32	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 2)
34	27	simpld 495	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 < 𝐴)
35	31, 30, 22, 33, 34	lttrd 11316	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐴)
36	22, 35	elrpd 12954	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
37		pilem2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (sin‘𝐴) = 0)
38		pilem1 25810	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐴 ∈ ℝ⁺ ∧ (sin‘𝐴) = 0))
39	36, 37, 38	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
40	39	ne0d 4295	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅)
41		infrecl 12137	. . . . . . . . . . . . . 14 ⊢ (((ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ ∧ (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦) → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ∈ ℝ)
42	4, 13, 41	mp3an13 1452	. . . . . . . . . . . . 13 ⊢ ((ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅ → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ∈ ℝ)
43	40, 42	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ∈ ℝ)
44		pilem1 25810	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0))
45		rpre 12923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
46	45	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
47		letric 11255	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
48	15, 46, 47	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
49	48	ord 862	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (¬ 2 ≤ 𝑥 → 𝑥 ≤ 2))
50	45	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ ℝ)
51		rpgt0 12927	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ⁺ → 0 < 𝑥)
52	51	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < 𝑥)
53		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ≤ 2)
54		0xr 11202	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℝ^*
55		elioc2 13327	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℝ^* ∧ 2 ∈ ℝ) → (𝑥 ∈ (0(,]2) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 ≤ 2)))
56	54, 15, 55	mp2an 690	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (0(,]2) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 ≤ 2))
57	50, 52, 53, 56	syl3anbrc 1343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ (0(,]2))
58		sin02gt0 16074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0(,]2) → 0 < (sin‘𝑥))
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < (sin‘𝑥))
60	59	gt0ne0d 11719	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → (sin‘𝑥) ≠ 0)
61	60	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ≤ 2 → (sin‘𝑥) ≠ 0))
62	49, 61	syld 47	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (¬ 2 ≤ 𝑥 → (sin‘𝑥) ≠ 0))
63	62	necon4bd 2963	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((sin‘𝑥) = 0 → 2 ≤ 𝑥))
64	63	expimpd 454	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0) → 2 ≤ 𝑥))
65	44, 64	biimtrid 241	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 2 ≤ 𝑥))
66	65	ralrimiv 3142	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥)
67		infregelb 12139	. . . . . . . . . . . . . 14 ⊢ ((((ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ ∧ (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦) ∧ 2 ∈ ℝ) → (2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ↔ ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥))
68	5, 40, 14, 30, 67	syl31anc 1373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ↔ ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥))
69	66, 68	mpbird 256	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ))
70		pilem2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (sin‘𝐵) = 0)
71		pilem1 25810	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐵 ∈ ℝ⁺ ∧ (sin‘𝐵) = 0))
72	16, 70, 71	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
73		infrelb 12140	. . . . . . . . . . . . 13 ⊢ (((ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ∧ 𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))) → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ 𝐵)
74	5, 14, 72, 73	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ 𝐵)
75	30, 43, 17, 69, 74	letrd 11312	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝐵)
76	15, 32	pm3.2i 471	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ ∧ 0 < 2)
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
78		lemul2 12008	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
79	30, 17, 77, 78	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
80	75, 79	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 2) ≤ (2 · 𝐵))
81	29, 80	eqbrtrrid 5141	. . . . . . . . 9 ⊢ (𝜑 → 4 ≤ (2 · 𝐵))
82	22, 25, 19, 28, 81	ltletrd 11315	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < (2 · 𝐵))
83	22, 19	posdifd 11742	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < (2 · 𝐵) ↔ 0 < ((2 · 𝐵) − 𝐴)))
84	82, 83	mpbid 231	. . . . . . 7 ⊢ (𝜑 → 0 < ((2 · 𝐵) − 𝐴))
85	23, 84	elrpd 12954	. . . . . 6 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ⁺)
86	19	recnd 11183	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℂ)
87	22	recnd 11183	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
88		sinsub 16050	. . . . . . . 8 ⊢ (((2 · 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (sin‘((2 · 𝐵) − 𝐴)) = (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))))
89	86, 87, 88	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))))
90	17	recnd 11183	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
91		sin2t 16059	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℂ → (sin‘(2 · 𝐵)) = (2 · ((sin‘𝐵) · (cos‘𝐵))))
92	90, 91	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (sin‘(2 · 𝐵)) = (2 · ((sin‘𝐵) · (cos‘𝐵))))
93	70	oveq1d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = (0 · (cos‘𝐵)))
94	90	coscld 16013	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (cos‘𝐵) ∈ ℂ)
95	94	mul02d 11353	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 · (cos‘𝐵)) = 0)
96	93, 95	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = 0)
97	96	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = (2 · 0))
98		2t0e0 12322	. . . . . . . . . . . . 13 ⊢ (2 · 0) = 0
99	97, 98	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = 0)
100	92, 99	eqtrd 2776	. . . . . . . . . . 11 ⊢ (𝜑 → (sin‘(2 · 𝐵)) = 0)
101	100	oveq1d 7372	. . . . . . . . . 10 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = (0 · (cos‘𝐴)))
102	87	coscld 16013	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
103	102	mul02d 11353	. . . . . . . . . 10 ⊢ (𝜑 → (0 · (cos‘𝐴)) = 0)
104	101, 103	eqtrd 2776	. . . . . . . . 9 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = 0)
105	37	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = ((cos‘(2 · 𝐵)) · 0))
106	86	coscld 16013	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘(2 · 𝐵)) ∈ ℂ)
107	106	mul01d 11354	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · 0) = 0)
108	105, 107	eqtrd 2776	. . . . . . . . 9 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = 0)
109	104, 108	oveq12d 7375	. . . . . . . 8 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = (0 − 0))
110		0m0e0 12273	. . . . . . . 8 ⊢ (0 − 0) = 0
111	109, 110	eqtrdi 2792	. . . . . . 7 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = 0)
112	89, 111	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = 0)
113		pilem1 25810	. . . . . 6 ⊢ (((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (((2 · 𝐵) − 𝐴) ∈ ℝ⁺ ∧ (sin‘((2 · 𝐵) − 𝐴)) = 0))
114	85, 112, 113	sylanbrc 583	. . . . 5 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
115		infrelb 12140	. . . . 5 ⊢ (((ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ∧ ((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0}))) → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ ((2 · 𝐵) − 𝐴))
116	5, 14, 114, 115	syl3anc 1371	. . . 4 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ ((2 · 𝐵) − 𝐴))
117	1, 116	eqbrtrid 5140	. . 3 ⊢ (𝜑 → π ≤ ((2 · 𝐵) − 𝐴))
118	1, 43	eqeltrid 2842	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
119		leaddsub 11631	. . . 4 ⊢ ((π ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 · 𝐵) ∈ ℝ) → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
120	118, 22, 19, 119	syl3anc 1371	. . 3 ⊢ (𝜑 → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
121	117, 120	mpbird 256	. 2 ⊢ (𝜑 → (π + 𝐴) ≤ (2 · 𝐵))
122	118, 22	readdcld 11184	. . 3 ⊢ (𝜑 → (π + 𝐴) ∈ ℝ)
123		ledivmul 12031	. . 3 ⊢ (((π + 𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
124	122, 17, 77, 123	syl3anc 1371	. 2 ⊢ (𝜑 → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
125	121, 124	mpbird 256	1 ⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 {csn 4586 class class class wbr 5105 ^◡ccnv 5632 “ cima 5636 ‘cfv 6496 (class class class)co 7357 infcinf 9377 ℂcc 11049 ℝcr 11050 0cc0 11051 + caddc 11054 · cmul 11056 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 2c2 12208 4c4 12210 ℝ⁺crp 12915 (,)cioo 13264 (,]cioc 13265 sincsin 15946 cosccos 15947 πcpi 15949

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-ioo 13268 df-ioc 13269 df-ico 13270 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955

This theorem is referenced by: pilem3 25812

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