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Theorem pilem2 25042

Description: Lemma for pire 25046, pigt2lt4 25044 and sinpi 25045. (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 15428	. . . 4 ⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
2		inss1 4207	. . . . . . 7 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ⁺
3		rpssre 12399	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
4	2, 3	sstri 3978	. . . . . 6 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ)
6		0re 10645	. . . . . . 7 ⊢ 0 ∈ ℝ
7		elinel1 4174	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 𝑦 ∈ ℝ⁺)
8	7	rpge0d 12438	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 0 ≤ 𝑦)
9	8	rgen 3150	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦
10		breq1 5071	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
11	10	ralbidv 3199	. . . . . . . 8 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦))
12	11	rspcev 3625	. . . . . . 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 11714	. . . . . . . . 9 ⊢ 2 ∈ ℝ
16		pilem2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
17	16	rpred 12434	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
18		remulcl 10624	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 · 𝐵) ∈ ℝ)
19	15, 17, 18	sylancr 589	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
20		pilem2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (2(,)4))
21		elioore 12771	. . . . . . . . 9 ⊢ (𝐴 ∈ (2(,)4) → 𝐴 ∈ ℝ)
22	20, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
23	19, 22	resubcld 11070	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ)
24		4re 11724	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ∈ ℝ)
26		eliooord 12799	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (2(,)4) → (2 < 𝐴 ∧ 𝐴 < 4))
27	20, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝐴 ∧ 𝐴 < 4))
28	27	simprd 498	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 4)
29		2t2e4 11804	. . . . . . . . . 10 ⊢ (2 · 2) = 4
30	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ)
31		0red 10646	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ∈ ℝ)
32		2pos 11743	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
33	32	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 2)
34	27	simpld 497	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 < 𝐴)
35	31, 30, 22, 33, 34	lttrd 10803	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐴)
36	22, 35	elrpd 12431	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
37		pilem2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (sin‘𝐴) = 0)
38		pilem1 25041	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐴 ∈ ℝ⁺ ∧ (sin‘𝐴) = 0))
39	36, 37, 38	sylanbrc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
40	39	ne0d 4303	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅)
41		infrecl 11625	. . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . 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 25041	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0))
45		rpre 12400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
46	45	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
47		letric 10742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
48	15, 46, 47	sylancr 589	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
49	48	ord 860	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (¬ 2 ≤ 𝑥 → 𝑥 ≤ 2))
50	45	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ ℝ)
51		rpgt0 12404	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ⁺ → 0 < 𝑥)
52	51	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < 𝑥)
53		simpr 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ≤ 2)
54		0xr 10690	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℝ^*
55		elioc2 12802	. . . . . . . . . . . . . . . . . . . . . . 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 1339	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ (0(,]2))
58		sin02gt0 15547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0(,]2) → 0 < (sin‘𝑥))
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < (sin‘𝑥))
60	59	gt0ne0d 11206	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → (sin‘𝑥) ≠ 0)
61	60	ex 415	. . . . . . . . . . . . . . . . . 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 3038	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((sin‘𝑥) = 0 → 2 ≤ 𝑥))
64	63	expimpd 456	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0) → 2 ≤ 𝑥))
65	44, 64	syl5bi 244	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 2 ≤ 𝑥))
66	65	ralrimiv 3183	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥)
67		infregelb 11627	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ↔ ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥))
69	66, 68	mpbird 259	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ))
70		pilem2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (sin‘𝐵) = 0)
71		pilem1 25041	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐵 ∈ ℝ⁺ ∧ (sin‘𝐵) = 0))
72	16, 70, 71	sylanbrc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
73		infrelb 11628	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 12 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ 𝐵)
75	30, 43, 17, 69, 74	letrd 10799	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝐵)
76	15, 32	pm3.2i 473	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ ∧ 0 < 2)
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
78		lemul2 11495	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
79	30, 17, 77, 78	syl3anc 1367	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
80	75, 79	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 2) ≤ (2 · 𝐵))
81	29, 80	eqbrtrrid 5104	. . . . . . . . 9 ⊢ (𝜑 → 4 ≤ (2 · 𝐵))
82	22, 25, 19, 28, 81	ltletrd 10802	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < (2 · 𝐵))
83	22, 19	posdifd 11229	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < (2 · 𝐵) ↔ 0 < ((2 · 𝐵) − 𝐴)))
84	82, 83	mpbid 234	. . . . . . 7 ⊢ (𝜑 → 0 < ((2 · 𝐵) − 𝐴))
85	23, 84	elrpd 12431	. . . . . 6 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ⁺)
86	19	recnd 10671	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℂ)
87	22	recnd 10671	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
88		sinsub 15523	. . . . . . . 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 586	. . . . . . 7 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))))
90	17	recnd 10671	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
91		sin2t 15532	. . . . . . . . . . . . 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 7173	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = (0 · (cos‘𝐵)))
94	90	coscld 15486	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (cos‘𝐵) ∈ ℂ)
95	94	mul02d 10840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 · (cos‘𝐵)) = 0)
96	93, 95	eqtrd 2858	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = 0)
97	96	oveq2d 7174	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = (2 · 0))
98		2t0e0 11809	. . . . . . . . . . . . 13 ⊢ (2 · 0) = 0
99	97, 98	syl6eq 2874	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = 0)
100	92, 99	eqtrd 2858	. . . . . . . . . . 11 ⊢ (𝜑 → (sin‘(2 · 𝐵)) = 0)
101	100	oveq1d 7173	. . . . . . . . . 10 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = (0 · (cos‘𝐴)))
102	87	coscld 15486	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
103	102	mul02d 10840	. . . . . . . . . 10 ⊢ (𝜑 → (0 · (cos‘𝐴)) = 0)
104	101, 103	eqtrd 2858	. . . . . . . . 9 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = 0)
105	37	oveq2d 7174	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = ((cos‘(2 · 𝐵)) · 0))
106	86	coscld 15486	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘(2 · 𝐵)) ∈ ℂ)
107	106	mul01d 10841	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · 0) = 0)
108	105, 107	eqtrd 2858	. . . . . . . . 9 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = 0)
109	104, 108	oveq12d 7176	. . . . . . . 8 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = (0 − 0))
110		0m0e0 11760	. . . . . . . 8 ⊢ (0 − 0) = 0
111	109, 110	syl6eq 2874	. . . . . . 7 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = 0)
112	89, 111	eqtrd 2858	. . . . . 6 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = 0)
113		pilem1 25041	. . . . . 6 ⊢ (((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (((2 · 𝐵) − 𝐴) ∈ ℝ⁺ ∧ (sin‘((2 · 𝐵) − 𝐴)) = 0))
114	85, 112, 113	sylanbrc 585	. . . . 5 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
115		infrelb 11628	. . . . 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 1367	. . . 4 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ ((2 · 𝐵) − 𝐴))
117	1, 116	eqbrtrid 5103	. . 3 ⊢ (𝜑 → π ≤ ((2 · 𝐵) − 𝐴))
118	1, 43	eqeltrid 2919	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
119		leaddsub 11118	. . . 4 ⊢ ((π ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 · 𝐵) ∈ ℝ) → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
120	118, 22, 19, 119	syl3anc 1367	. . 3 ⊢ (𝜑 → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
121	117, 120	mpbird 259	. 2 ⊢ (𝜑 → (π + 𝐴) ≤ (2 · 𝐵))
122	118, 22	readdcld 10672	. . 3 ⊢ (𝜑 → (π + 𝐴) ∈ ℝ)
123		ledivmul 11518	. . 3 ⊢ (((π + 𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
124	122, 17, 77, 123	syl3anc 1367	. 2 ⊢ (𝜑 → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
125	121, 124	mpbird 259	1 ⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 class class class wbr 5068 ^◡ccnv 5556 “ cima 5560 ‘cfv 6357 (class class class)co 7158 infcinf 8907 ℂcc 10537 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 ℝ^*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 2c2 11695 4c4 11697 ℝ⁺crp 12392 (,)cioo 12741 (,]cioc 12742 sincsin 15419 cosccos 15420 πcpi 15422

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ioo 12745 df-ioc 12746 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428

This theorem is referenced by: pilem3 25043

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