pilem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pilem2		Structured version Visualization version GIF version

Theorem pilem2 25956

Description: Lemma for pire 25960, pigt2lt4 25958 and sinpi 25959. (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 16013	. . . 4 ⊢ π = inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < )
2		inss1 4228	. . . . . . 7 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ⁺
3		rpssre 12978	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
4	2, 3	sstri 3991	. . . . . 6 ⊢ (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ⊆ ℝ)
6		0re 11213	. . . . . . 7 ⊢ 0 ∈ ℝ
7		elinel1 4195	. . . . . . . . 9 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 𝑦 ∈ ℝ⁺)
8	7	rpge0d 13017	. . . . . . . 8 ⊢ (𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) → 0 ≤ 𝑦)
9	8	rgen 3064	. . . . . . 7 ⊢ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦
10		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
11	10	ralbidv 3178	. . . . . . . 8 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦))
12	11	rspcev 3613	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))𝑥 ≤ 𝑦)
13	6, 9, 12	mp2an 691	. . . . . 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 12283	. . . . . . . . 9 ⊢ 2 ∈ ℝ
16		pilem2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
17	16	rpred 13013	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
18		remulcl 11192	. . . . . . . . 9 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 · 𝐵) ∈ ℝ)
19	15, 17, 18	sylancr 588	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℝ)
20		pilem2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (2(,)4))
21		elioore 13351	. . . . . . . . 9 ⊢ (𝐴 ∈ (2(,)4) → 𝐴 ∈ ℝ)
22	20, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
23	19, 22	resubcld 11639	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ)
24		4re 12293	. . . . . . . . . 10 ⊢ 4 ∈ ℝ
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 4 ∈ ℝ)
26		eliooord 13380	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (2(,)4) → (2 < 𝐴 ∧ 𝐴 < 4))
27	20, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (2 < 𝐴 ∧ 𝐴 < 4))
28	27	simprd 497	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 < 4)
29		2t2e4 12373	. . . . . . . . . 10 ⊢ (2 · 2) = 4
30	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℝ)
31		0red 11214	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ∈ ℝ)
32		2pos 12312	. . . . . . . . . . . . . . . . . 18 ⊢ 0 < 2
33	32	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 2)
34	27	simpld 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2 < 𝐴)
35	31, 30, 22, 33, 34	lttrd 11372	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 < 𝐴)
36	22, 35	elrpd 13010	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
37		pilem2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (sin‘𝐴) = 0)
38		pilem1 25955	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐴 ∈ ℝ⁺ ∧ (sin‘𝐴) = 0))
39	36, 37, 38	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
40	39	ne0d 4335	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ⁺ ∩ (^◡sin “ {0})) ≠ ∅)
41		infrecl 12193	. . . . . . . . . . . . . 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 1453	. . . . . . . . . . . . 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 25955	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝑥 ∈ ℝ⁺ ∧ (sin‘𝑥) = 0))
45		rpre 12979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
46	45	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
47		letric 11311	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
48	15, 46, 47	sylancr 588	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 ≤ 𝑥 ∨ 𝑥 ≤ 2))
49	48	ord 863	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (¬ 2 ≤ 𝑥 → 𝑥 ≤ 2))
50	45	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ ℝ)
51		rpgt0 12983	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ⁺ → 0 < 𝑥)
52	51	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < 𝑥)
53		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ≤ 2)
54		0xr 11258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℝ^*
55		elioc2 13384	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℝ^* ∧ 2 ∈ ℝ) → (𝑥 ∈ (0(,]2) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 ≤ 2)))
56	54, 15, 55	mp2an 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ (0(,]2) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 ≤ 2))
57	50, 52, 53, 56	syl3anbrc 1344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 𝑥 ∈ (0(,]2))
58		sin02gt0 16132	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (0(,]2) → 0 < (sin‘𝑥))
59	57, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → 0 < (sin‘𝑥))
60	59	gt0ne0d 11775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑥 ≤ 2) → (sin‘𝑥) ≠ 0)
61	60	ex 414	. . . . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((sin‘𝑥) = 0 → 2 ≤ 𝑥))
64	63	expimpd 455	. . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥)
67		infregelb 12195	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ↔ ∀𝑥 ∈ (ℝ⁺ ∩ (^◡sin “ {0}))2 ≤ 𝑥))
69	66, 68	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ))
70		pilem2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (sin‘𝐵) = 0)
71		pilem1 25955	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (𝐵 ∈ ℝ⁺ ∧ (sin‘𝐵) = 0))
72	16, 70, 71	sylanbrc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
73		infrelb 12196	. . . . . . . . . . . . 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 1372	. . . . . . . . . . . 12 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ 𝐵)
75	30, 43, 17, 69, 74	letrd 11368	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≤ 𝐵)
76	15, 32	pm3.2i 472	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℝ ∧ 0 < 2)
77	76	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 ∈ ℝ ∧ 0 < 2))
78		lemul2 12064	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
79	30, 17, 77, 78	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → (2 ≤ 𝐵 ↔ (2 · 2) ≤ (2 · 𝐵)))
80	75, 79	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (2 · 2) ≤ (2 · 𝐵))
81	29, 80	eqbrtrrid 5184	. . . . . . . . 9 ⊢ (𝜑 → 4 ≤ (2 · 𝐵))
82	22, 25, 19, 28, 81	ltletrd 11371	. . . . . . . 8 ⊢ (𝜑 → 𝐴 < (2 · 𝐵))
83	22, 19	posdifd 11798	. . . . . . . 8 ⊢ (𝜑 → (𝐴 < (2 · 𝐵) ↔ 0 < ((2 · 𝐵) − 𝐴)))
84	82, 83	mpbid 231	. . . . . . 7 ⊢ (𝜑 → 0 < ((2 · 𝐵) − 𝐴))
85	23, 84	elrpd 13010	. . . . . 6 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ ℝ⁺)
86	19	recnd 11239	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐵) ∈ ℂ)
87	22	recnd 11239	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
88		sinsub 16108	. . . . . . . 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 585	. . . . . . 7 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))))
90	17	recnd 11239	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
91		sin2t 16117	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = (0 · (cos‘𝐵)))
94	90	coscld 16071	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (cos‘𝐵) ∈ ℂ)
95	94	mul02d 11409	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0 · (cos‘𝐵)) = 0)
96	93, 95	eqtrd 2773	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((sin‘𝐵) · (cos‘𝐵)) = 0)
97	96	oveq2d 7422	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = (2 · 0))
98		2t0e0 12378	. . . . . . . . . . . . 13 ⊢ (2 · 0) = 0
99	97, 98	eqtrdi 2789	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · ((sin‘𝐵) · (cos‘𝐵))) = 0)
100	92, 99	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝜑 → (sin‘(2 · 𝐵)) = 0)
101	100	oveq1d 7421	. . . . . . . . . 10 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = (0 · (cos‘𝐴)))
102	87	coscld 16071	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘𝐴) ∈ ℂ)
103	102	mul02d 11409	. . . . . . . . . 10 ⊢ (𝜑 → (0 · (cos‘𝐴)) = 0)
104	101, 103	eqtrd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((sin‘(2 · 𝐵)) · (cos‘𝐴)) = 0)
105	37	oveq2d 7422	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = ((cos‘(2 · 𝐵)) · 0))
106	86	coscld 16071	. . . . . . . . . . 11 ⊢ (𝜑 → (cos‘(2 · 𝐵)) ∈ ℂ)
107	106	mul01d 11410	. . . . . . . . . 10 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · 0) = 0)
108	105, 107	eqtrd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((cos‘(2 · 𝐵)) · (sin‘𝐴)) = 0)
109	104, 108	oveq12d 7424	. . . . . . . 8 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = (0 − 0))
110		0m0e0 12329	. . . . . . . 8 ⊢ (0 − 0) = 0
111	109, 110	eqtrdi 2789	. . . . . . 7 ⊢ (𝜑 → (((sin‘(2 · 𝐵)) · (cos‘𝐴)) − ((cos‘(2 · 𝐵)) · (sin‘𝐴))) = 0)
112	89, 111	eqtrd 2773	. . . . . 6 ⊢ (𝜑 → (sin‘((2 · 𝐵) − 𝐴)) = 0)
113		pilem1 25955	. . . . . 6 ⊢ (((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})) ↔ (((2 · 𝐵) − 𝐴) ∈ ℝ⁺ ∧ (sin‘((2 · 𝐵) − 𝐴)) = 0))
114	85, 112, 113	sylanbrc 584	. . . . 5 ⊢ (𝜑 → ((2 · 𝐵) − 𝐴) ∈ (ℝ⁺ ∩ (^◡sin “ {0})))
115		infrelb 12196	. . . . 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 1372	. . . 4 ⊢ (𝜑 → inf((ℝ⁺ ∩ (^◡sin “ {0})), ℝ, < ) ≤ ((2 · 𝐵) − 𝐴))
117	1, 116	eqbrtrid 5183	. . 3 ⊢ (𝜑 → π ≤ ((2 · 𝐵) − 𝐴))
118	1, 43	eqeltrid 2838	. . . 4 ⊢ (𝜑 → π ∈ ℝ)
119		leaddsub 11687	. . . 4 ⊢ ((π ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (2 · 𝐵) ∈ ℝ) → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
120	118, 22, 19, 119	syl3anc 1372	. . 3 ⊢ (𝜑 → ((π + 𝐴) ≤ (2 · 𝐵) ↔ π ≤ ((2 · 𝐵) − 𝐴)))
121	117, 120	mpbird 257	. 2 ⊢ (𝜑 → (π + 𝐴) ≤ (2 · 𝐵))
122	118, 22	readdcld 11240	. . 3 ⊢ (𝜑 → (π + 𝐴) ∈ ℝ)
123		ledivmul 12087	. . 3 ⊢ (((π + 𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
124	122, 17, 77, 123	syl3anc 1372	. 2 ⊢ (𝜑 → (((π + 𝐴) / 2) ≤ 𝐵 ↔ (π + 𝐴) ≤ (2 · 𝐵)))
125	121, 124	mpbird 257	1 ⊢ (𝜑 → ((π + 𝐴) / 2) ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 class class class wbr 5148 ^◡ccnv 5675 “ cima 5679 ‘cfv 6541 (class class class)co 7406 infcinf 9433 ℂcc 11105 ℝcr 11106 0cc0 11107 + caddc 11110 · cmul 11112 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 − cmin 11441 / cdiv 11868 2c2 12264 4c4 12266 ℝ⁺crp 12971 (,)cioo 13321 (,]cioc 13322 sincsin 16004 cosccos 16005 πcpi 16007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-ioo 13325 df-ioc 13326 df-ico 13327 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-ef 16008 df-sin 16010 df-cos 16011 df-pi 16013

This theorem is referenced by: pilem3 25957

W3C validator