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Theorem pilem2 24251

Description: Lemma for pire 24255, pigt2lt4 24253 and sinpi 24254. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by AV, 14-Sep-2020.)

**Hypotheses**
Ref	Expression
pilem.1	⊢ (𝜑 → 𝐴 ∈ (2(,)4))
pilem.2	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
pilem.3	⊢ (𝜑 → (sin‘𝐴) = 0)
pilem.4	⊢ (𝜑 → (sin‘𝐵) = 0)
pilem.5	⊢ (𝜑 → π < 𝐴)

**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 14847	. . . 4 ⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
2		inss1 3866	. . . . . . 7 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ⁺
3		rpssre 11881	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
4	2, 3	sstri 3645	. . . . . 6 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ)
6		0re 10078	. . . . . . 7 ⊢ 0 ∈ ℝ
7	2	sseli 3632	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 𝑦 ∈ ℝ⁺)
8	7	rpge0d 11914	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 0 ≤ 𝑦)
9	8	rgen 2951	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦
10		breq1 4688	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
11	10	ralbidv 3015	. . . . . . . 8 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦))
12	11	rspcev 3340	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦)
13	6, 9, 12	mp2an 708	. . . . . 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 11128	. . . . . . . . 9 ⊢ 2 ∈ ℝ
16		pilem.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
17	16	rpred 11910	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
18		remulcl 10059	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 · 𝐵) ∈ ℝ)
19	15, 17, 18	sylancr 696	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
20		pilem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (2(,)4))
21		elioore 12243	. . . . . . . . 9 ⊢ (𝐴 ∈ (2(,)4) → 𝐴 ∈ ℝ)
22	20, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
23	19, 22	resubcld 10496	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ)
24		4re 11135	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ∈ ℝ)
26		eliooord 12271	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (2(,)4) → (2 < 𝐴 ∧ 𝐴 < 4))
27	20, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝐴 ∧ 𝐴 < 4))
28	27	simprd 478	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 4)
29		2t2e4 11215	. . . . . . . . . 10 ⊢ (2 · 2) = 4
30	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ)
31		0red 10079	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ∈ ℝ)
32		2pos 11150	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
33	32	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 2)
34	27	simpld 474	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 < 𝐴)
35	31, 30, 22, 33, 34	lttrd 10236	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐴)
36	22, 35	elrpd 11907	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
37		pilem.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (sin‘𝐴) = 0)
38		pilem1 24250	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐴 ∈ ℝ⁺ ∧ (sin‘𝐴) = 0))
39	36, 37, 38	sylanbrc 699	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
40		ne0i 3954	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅)
41	39, 40	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅)
42		infrecl 11043	. . . . . . . . . . . . . 14 ⊢ (((ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ ∧ (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦) → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ∈ ℝ)
43	4, 13, 42	mp3an13 1455	. . . . . . . . . . . . 13 ⊢ ((ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅ → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ∈ ℝ)
44	41, 43	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ∈ ℝ)
45		pilem1 24250	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0))
46		rpre 11877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
47	46	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
48		letric 10175	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
49	15, 47, 48	sylancr 696	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
50	49	ord 391	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (¬ 2 ≤ 𝑥 → 𝑥 ≤ 2))
51	46	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ ℝ)
52		rpgt0 11882	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ⁺ → 0 < 𝑥)
53	52	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < 𝑥)
54		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ≤ 2)
55		0xr 10124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℝ^*
56		elioc2 12274	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℝ^* ∧ 2 ∈ ℝ) → (𝑥 ∈ (0(,]2) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 ≤ 2)))
57	55, 15, 56	mp2an 708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (0(,]2) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 ≤ 2))
58	51, 53, 54, 57	syl3anbrc 1265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ (0(,]2))
59		sin02gt0 14966	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0(,]2) → 0 < (sin‘𝑥))
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < (sin‘𝑥))
61	60	gt0ne0d 10630	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → (sin‘𝑥) ≠ 0)
62	61	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ≤ 2 → (sin‘𝑥) ≠ 0))
63	50, 62	syld 47	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (¬ 2 ≤ 𝑥 → (sin‘𝑥) ≠ 0))
64	63	necon4bd 2843	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((sin‘𝑥) = 0 → 2 ≤ 𝑥))
65	64	expimpd 628	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0) → 2 ≤ 𝑥))
66	45, 65	syl5bi 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 2 ≤ 𝑥))
67	66	ralrimiv 2994	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥)
68		infregelb 11045	. . . . . . . . . . . . . 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))
69	5, 41, 14, 30, 68	syl31anc 1369	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ↔ ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥))
70	67, 69	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ))
71		pilem.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (sin‘𝐵) = 0)
72		pilem1 24250	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐵 ∈ ℝ⁺ ∧ (sin‘𝐵) = 0))
73	16, 71, 72	sylanbrc 699	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
74		infrelb 11046	. . . . . . . . . . . . 13 ⊢ (((ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ∧ 𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))) → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ 𝐵)
75	5, 14, 73, 74	syl3anc 1366	. . . . . . . . . . . 12 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ 𝐵)
76	30, 44, 17, 70, 75	letrd 10232	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝐵)
77	15, 32	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ ∧ 0 < 2)
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
79		lemul2 10914	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
80	30, 17, 78, 79	syl3anc 1366	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
81	76, 80	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 2) ≤ (2 · 𝐵))
82	29, 81	syl5eqbrr 4721	. . . . . . . . 9 ⊢ (𝜑 → 4 ≤ (2 · 𝐵))
83	22, 25, 19, 28, 82	ltletrd 10235	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < (2 · 𝐵))
84	22, 19	posdifd 10652	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < (2 · 𝐵) ↔ 0 < ((2 · 𝐵) − 𝐴)))
85	83, 84	mpbid 222	. . . . . . 7 ⊢ (𝜑 → 0 < ((2 · 𝐵) − 𝐴))
86	23, 85	elrpd 11907	. . . . . 6 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ⁺)
87	19	recnd 10106	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℂ)
88	22	recnd 10106	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
89		sinsub 14942	. . . . . . . 8 ⊢ (((2 · 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℂ) → (sin‘((2 · 𝐵) − 𝐴)) = (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))))
90	87, 88, 89	syl2anc 694	. . . . . . 7 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))))
91	17	recnd 10106	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
92		sin2t 14951	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℂ → (sin‘(2 · 𝐵)) = (2 · ((sin‘𝐵) · (cos‘𝐵))))
93	91, 92	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (sin‘(2 · 𝐵)) = (2 · ((sin‘𝐵) · (cos‘𝐵))))
94	71	oveq1d 6705	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = (0 · (cos‘𝐵)))
95	91	coscld 14905	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (cos‘𝐵) ∈ ℂ)
96	95	mul02d 10272	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 · (cos‘𝐵)) = 0)
97	94, 96	eqtrd 2685	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = 0)
98	97	oveq2d 6706	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = (2 · 0))
99		2t0e0 11221	. . . . . . . . . . . . 13 ⊢ (2 · 0) = 0
100	98, 99	syl6eq 2701	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = 0)
101	93, 100	eqtrd 2685	. . . . . . . . . . 11 ⊢ (𝜑 → (sin‘(2 · 𝐵)) = 0)
102	101	oveq1d 6705	. . . . . . . . . 10 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = (0 · (cos‘𝐴)))
103	88	coscld 14905	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
104	103	mul02d 10272	. . . . . . . . . 10 ⊢ (𝜑 → (0 · (cos‘𝐴)) = 0)
105	102, 104	eqtrd 2685	. . . . . . . . 9 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = 0)
106	37	oveq2d 6706	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = ((cos‘(2 · 𝐵)) · 0))
107	87	coscld 14905	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘(2 · 𝐵)) ∈ ℂ)
108	107	mul01d 10273	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · 0) = 0)
109	106, 108	eqtrd 2685	. . . . . . . . 9 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = 0)
110	105, 109	oveq12d 6708	. . . . . . . 8 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = (0 − 0))
111		0m0e0 11168	. . . . . . . 8 ⊢ (0 − 0) = 0
112	110, 111	syl6eq 2701	. . . . . . 7 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = 0)
113	90, 112	eqtrd 2685	. . . . . 6 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = 0)
114		pilem1 24250	. . . . . 6 ⊢ (((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (((2 · 𝐵) − 𝐴) ∈ ℝ⁺ ∧ (sin‘((2 · 𝐵) − 𝐴)) = 0))
115	86, 113, 114	sylanbrc 699	. . . . 5 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
116		infrelb 11046	. . . . 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))
117	5, 14, 115, 116	syl3anc 1366	. . . 4 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ ((2 · 𝐵) − 𝐴))
118	1, 117	syl5eqbr 4720	. . 3 ⊢ (𝜑 → π ≤ ((2 · 𝐵) − 𝐴))
119	1, 44	syl5eqel 2734	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
120		leaddsub 10542	. . . 4 ⊢ ((π ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 · 𝐵) ∈ ℝ) → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
121	119, 22, 19, 120	syl3anc 1366	. . 3 ⊢ (𝜑 → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
122	118, 121	mpbird 247	. 2 ⊢ (𝜑 → (π + 𝐴) ≤ (2 · 𝐵))
123	119, 22	readdcld 10107	. . 3 ⊢ (𝜑 → (π + 𝐴) ∈ ℝ)
124		ledivmul 10937	. . 3 ⊢ (((π + 𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
125	123, 17, 78, 124	syl3anc 1366	. 2 ⊢ (𝜑 → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
126	122, 125	mpbird 247	1 ⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 382 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 {csn 4210 class class class wbr 4685 ^◡ccnv 5142 “ cima 5146 ‘cfv 5926 (class class class)co 6690 infcinf 8388 ℂcc 9972 ℝcr 9973 0cc0 9974 + caddc 9977 · cmul 9979 ℝ^*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 / cdiv 10722 2c2 11108 4c4 11110 ℝ⁺crp 11870 (,)cioo 12213 (,]cioc 12214 sincsin 14838 cosccos 14839 πcpi 14841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-ioo 12217 df-ioc 12218 df-ico 12219 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847

This theorem is referenced by: pilem3 24252

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