Theorem cdj1i 28676

Description: Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdj1.1	⊢ 𝐴 ∈ Sℋ
cdj1.2	⊢ 𝐵 ∈ Sℋ

Assertion

Ref	Expression
cdj1i	⊢ (∃𝑤 ∈ ℝ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝑣,𝐵,𝑦,𝑧,𝑤

Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem cdj1i

Step	Hyp	Ref	Expression
1		gt0ne0 10372	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 𝑤 ≠ 0)
2		rereccl 10622	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℝ)
3	1, 2	syldan 486	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → (1 / 𝑤) ∈ ℝ)
4	3	adantrr 749	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → (1 / 𝑤) ∈ ℝ)
5		recgt0 10746	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 0 < (1 / 𝑤))
6	5	adantrr 749	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → 0 < (1 / 𝑤))
7		1red 9934	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → 1 ∈ ℝ)
8		1re 9918	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
9		neg1cn 11001	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℂ
10		cdj1.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ Sℋ
11	10	sheli 27455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ℋ)
12		hvmulcl 27254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℋ) → (-1 ·ℎ 𝑧) ∈ ℋ)
13	9, 11, 12	sylancr 694	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ ℋ)
14		normcl 27366	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-1 ·ℎ 𝑧) ∈ ℋ → (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐵 → (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ)
16	15	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ)
17		readdcl 9898	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
18	8, 16, 17	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
19	18	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
20		cdj1.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐴 ∈ Sℋ
21	20	sheli 27455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
22		hvsubcl 27258	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) ∈ ℋ)
23	21, 11, 22	syl2an 493	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) ∈ ℋ)
24		normcl 27366	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 −ℎ 𝑧) ∈ ℋ → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ)
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ)
26		remulcl 9900	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
27	25, 26	sylan2 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
28	27	anassrs 678	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
29	28	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
30		normge0 27367	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-1 ·ℎ 𝑧) ∈ ℋ → 0 ≤ (normℎ‘(-1 ·ℎ 𝑧)))
31	13, 30	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝐵 → 0 ≤ (normℎ‘(-1 ·ℎ 𝑧)))
32		addge01 10417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
33	8, 32	mpan 702	. . . . . . . . . . . . . . . . . . 19 ⊢ ((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
34	33	biimpa 500	. . . . . . . . . . . . . . . . . 18 ⊢ (((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ ∧ 0 ≤ (normℎ‘(-1 ·ℎ 𝑧))) → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
35	15, 31, 34	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
36	35	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
37		shmulcl 27459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝐵) → (-1 ·ℎ 𝑧) ∈ 𝐵)
38	10, 9, 37	mp3an12 1406	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ 𝐵)
39		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘𝑣) = (normℎ‘(-1 ·ℎ 𝑧)))
40	39	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → ((normℎ‘𝑦) + (normℎ‘𝑣)) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
41		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑦 +ℎ 𝑣) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
42	41	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘(𝑦 +ℎ 𝑣)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
43	42	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
44	40, 43	breq12d 4596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ↔ ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))))
45	44	rspcv 3278	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-1 ·ℎ 𝑧) ∈ 𝐵 → (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) → ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))))
46	38, 45	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝐵 → (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) → ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))))
47	46	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐵 ∧ ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
48	47	ad2ant2lr 780	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
49		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 = (normℎ‘𝑦) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
50	49	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ ((normℎ‘𝑦) = 1 → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
51	50	ad2antll 761	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
52		hvsubval 27257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
53	21, 11, 52	syl2an 493	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
54	53	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
55	54	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
56	55	adantll 746	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
57	56	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
58	48, 51, 57	3brtr4d 4615	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))))
59	7, 19, 29, 36, 58	letrd 10073	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → 1 ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))))
60	59	ex 449	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1) → 1 ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧)))))
61	60	adantllr 751	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1) → 1 ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧)))))
62		simplll 794	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℝ)
63	23	adantll 746	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) ∈ ℋ)
64	63, 24	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ)
65	62, 64, 26	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
66		simpllr 795	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 0 < 𝑤)
67		lediv1 10767	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ ∧ (𝑤 ∈ ℝ ∧ 0 < 𝑤)) → (1 ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ↔ (1 / 𝑤) ≤ ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤)))
68	8, 67	mp3an1 1403	. . . . . . . . . . . . . 14 ⊢ (((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ ∧ (𝑤 ∈ ℝ ∧ 0 < 𝑤)) → (1 ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ↔ (1 / 𝑤) ≤ ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤)))
69	65, 62, 66, 68	syl12anc 1316	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (1 ≤ (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ↔ (1 / 𝑤) ≤ ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤)))
70	61, 69	sylibd 228	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1) → (1 / 𝑤) ≤ ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤)))
71	70	imp 444	. . . . . . . . . . 11 ⊢ (((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (1 / 𝑤) ≤ ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤))
72	25	recnd 9947	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
73	72	adantll 746	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
74		recn 9905	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
75	74	ad3antrrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℂ)
76	1	ad2antrr 758	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ≠ 0)
77	73, 75, 76	divcan3d 10685	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤) = (normℎ‘(𝑦 −ℎ 𝑧)))
78	77	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤) = (normℎ‘(𝑦 −ℎ 𝑧)))
79	71, 78	breqtrd 4609	. . . . . . . . . 10 ⊢ (((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) ∧ (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) ∧ (normℎ‘𝑦) = 1)) → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))
80	79	exp43 638	. . . . . . . . 9 ⊢ (((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝐵 → (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) → ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))))
81	80	com23 84	. . . . . . . 8 ⊢ (((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) → (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) → (𝑧 ∈ 𝐵 → ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))))
82	81	ralrimdv 2951	. . . . . . 7 ⊢ (((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) → (∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) → ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
83	82	ralimdva 2945	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → (∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
84	83	impr 647	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))
85	4, 6, 84	jca32 556	. . . 4 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → ((1 / 𝑤) ∈ ℝ ∧ (0 < (1 / 𝑤) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))))
86	85	ex 449	. . 3 ⊢ (𝑤 ∈ ℝ → ((0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ((1 / 𝑤) ∈ ℝ ∧ (0 < (1 / 𝑤) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))))
87		breq2 4587	. . . . 5 ⊢ (𝑥 = (1 / 𝑤) → (0 < 𝑥 ↔ 0 < (1 / 𝑤)))
88		breq1 4586	. . . . . . 7 ⊢ (𝑥 = (1 / 𝑤) → (𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)) ↔ (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))
89	88	imbi2d 329	. . . . . 6 ⊢ (𝑥 = (1 / 𝑤) → (((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))) ↔ ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
90	89	2ralbidv 2972	. . . . 5 ⊢ (𝑥 = (1 / 𝑤) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
91	87, 90	anbi12d 743	. . . 4 ⊢ (𝑥 = (1 / 𝑤) → ((0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)))) ↔ (0 < (1 / 𝑤) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))))
92	91	rspcev 3282	. . 3 ⊢ (((1 / 𝑤) ∈ ℝ ∧ (0 < (1 / 𝑤) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
93	86, 92	syl6 34	. 2 ⊢ (𝑤 ∈ ℝ → ((0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))))))
94	93	rexlimiv 3009	1 ⊢ (∃𝑤 ∈ ℝ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954  -cneg 10146   / cdiv 10563   ℋchil 27160   +_ℎ cva 27161   ·_ℎ csm 27162  norm_ℎcno 27164   −_ℎ cmv 27166   S_ℋ csh 27169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-hilex 27240  ax-hfvadd 27241  ax-hv0cl 27244  ax-hfvmul 27246  ax-hvmul0 27251  ax-hfi 27320  ax-his1 27323  ax-his3 27325  ax-his4 27326

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-hnorm 27209  df-hvsub 27212  df-sh 27448

This theorem is referenced by: (None)