Theorem cdj1i 29864

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 10840	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 𝑤 ≠ 0)
2		rereccl 11093	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℝ)
3	1, 2	syldan 585	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → (1 / 𝑤) ∈ ℝ)
4	3	adantrr 707	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → (1 / 𝑤) ∈ ℝ)
5		recgt0 11221	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 0 < (1 / 𝑤))
6	5	adantrr 707	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → 0 < (1 / 𝑤))
7		1red 10377	. . . . . . . . . . . . . . . 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 10376	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
9		neg1cn 11496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℂ
10		cdj1.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ Sℋ
11	10	sheli 28643	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ℋ)
12		hvmulcl 28442	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℋ) → (-1 ·ℎ 𝑧) ∈ ℋ)
13	9, 11, 12	sylancr 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ ℋ)
14		normcl 28554	. . . . . . . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ)
17		readdcl 10355	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
18	8, 16, 17	sylancr 581	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
19	18	adantr 474	. . . . . . . . . . . . . . . 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 28643	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
22		hvsubcl 28446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) ∈ ℋ)
23	21, 11, 22	syl2an 589	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) ∈ ℋ)
24		normcl 28554	. . . . . . . . . . . . . . . . . . . 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 10357	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
27	25, 26	sylan2 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
28	27	anassrs 461	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
29	28	adantr 474	. . . . . . . . . . . . . . . 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 28555	. . . . . . . . . . . . . . . . . . 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 10885	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
33	8, 32	mpan 680	. . . . . . . . . . . . . . . . . . 19 ⊢ ((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
34	33	biimpa 470	. . . . . . . . . . . . . . . . . 18 ⊢ (((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ ∧ 0 ≤ (normℎ‘(-1 ·ℎ 𝑧))) → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
35	15, 31, 34	syl2anc 579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
36	35	ad2antlr 717	. . . . . . . . . . . . . . . 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 28647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝐵) → (-1 ·ℎ 𝑧) ∈ 𝐵)
38	10, 9, 37	mp3an12 1524	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ 𝐵)
39		fveq2 6446	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘𝑣) = (normℎ‘(-1 ·ℎ 𝑧)))
40	39	oveq2d 6938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → ((normℎ‘𝑦) + (normℎ‘𝑣)) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
41		oveq2 6930	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑦 +ℎ 𝑣) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
42	41	fveq2d 6450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘(𝑦 +ℎ 𝑣)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
43	42	oveq2d 6938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
44	40, 43	breq12d 4899	. . . . . . . . . . . . . . . . . . . . 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 3507	. . . . . . . . . . . . . . . . . . . 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 397	. . . . . . . . . . . . . . . . . 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 738	. . . . . . . . . . . . . . . . 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 6929	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 = (normℎ‘𝑦) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
50	49	eqcoms 2786	. . . . . . . . . . . . . . . . . 18 ⊢ ((normℎ‘𝑦) = 1 → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
51	50	ad2antll 719	. . . . . . . . . . . . . . . . 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 28445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
53	21, 11, 52	syl2an 589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
54	53	fveq2d 6450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
55	54	oveq2d 6938	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
56	55	adantll 704	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
57	56	adantr 474	. . . . . . . . . . . . . . . . 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 4918	. . . . . . . . . . . . . . . 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 10533	. . . . . . . . . . . . . . 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 403	. . . . . . . . . . . . . 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 709	. . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℝ)
63	23	adantll 704	. . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
66		simpllr 766	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 0 < 𝑤)
67		lediv1 11242	. . . . . . . . . . . . . . 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 1521	. . . . . . . . . . . . . 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 827	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 397	. . . . . . . . . . 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 10405	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
73	72	adantll 704	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
74		recn 10362	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
75	74	ad3antrrr 720	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℂ)
76	1	ad2antrr 716	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ≠ 0)
77	73, 75, 76	divcan3d 11156	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤) = (normℎ‘(𝑦 −ℎ 𝑧)))
78	77	adantr 474	. . . . . . . . . . 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 4912	. . . . . . . . . 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 429	. . . . . . . . 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 86	. . . . . . . 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 3150	. . . . . . 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 3144	. . . . . 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 448	. . . . 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 511	. . . 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 403	. . 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 4890	. . . . 5 ⊢ (𝑥 = (1 / 𝑤) → (0 < 𝑥 ↔ 0 < (1 / 𝑤)))
88		breq1 4889	. . . . . . 7 ⊢ (𝑥 = (1 / 𝑤) → (𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)) ↔ (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))
89	88	imbi2d 332	. . . . . 6 ⊢ (𝑥 = (1 / 𝑤) → (((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))) ↔ ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
90	89	2ralbidv 3171	. . . . 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 624	. . . 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 3511	. . 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 35	. 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 3209	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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  ∃wrex 3091   class class class wbr 4886  ‘cfv 6135  (class class class)co 6922  ℂcc 10270  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277   < clt 10411   ≤ cle 10412  -cneg 10607   / cdiv 11032   ℋchba 28348   +_ℎ cva 28349   ·_ℎ csm 28350  norm_ℎcno 28352   −_ℎ cmv 28354   S_ℋ csh 28357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-hilex 28428  ax-hfvadd 28429  ax-hv0cl 28432  ax-hfvmul 28434  ax-hvmul0 28439  ax-hfi 28508  ax-his1 28511  ax-his3 28513  ax-his4 28514

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-sup 8636  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-seq 13120  df-exp 13179  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-hnorm 28397  df-hvsub 28400  df-sh 28636

This theorem is referenced by: (None)