Theorem cdj1i 32529

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 11613	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 𝑤 ≠ 0)
2		rereccl 11871	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℝ)
3	1, 2	syldan 597	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → (1 / 𝑤) ∈ ℝ)
4	3	adantrr 723	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → (1 / 𝑤) ∈ ℝ)
5		recgt0 11999	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 0 < (1 / 𝑤))
6	5	adantrr 723	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → 0 < (1 / 𝑤))
7		1red 11143	. . . . . . . . . . . . . . . 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 11142	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
9		neg1cn 12142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℂ
10		cdj1.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ Sℋ
11	10	sheli 31310	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ℋ)
12		hvmulcl 31109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℋ) → (-1 ·ℎ 𝑧) ∈ ℋ)
13	9, 11, 12	sylancr 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ ℋ)
14		normcl 31221	. . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ)
17		readdcl 11119	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
18	8, 16, 17	sylancr 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
19	18	adantr 481	. . . . . . . . . . . . . . . 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 31310	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
22		hvsubcl 31113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) ∈ ℋ)
23	21, 11, 22	syl2an 602	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) ∈ ℋ)
24		normcl 31221	. . . . . . . . . . . . . . . . . . . 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 11121	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
27	25, 26	sylan2 599	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
28	27	anassrs 468	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
29	28	adantr 481	. . . . . . . . . . . . . . . 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 31222	. . . . . . . . . . . . . . . . . . 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 11658	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
33	8, 32	mpan 696	. . . . . . . . . . . . . . . . . . 19 ⊢ ((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
34	33	biimpa 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ ∧ 0 ≤ (normℎ‘(-1 ·ℎ 𝑧))) → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
35	15, 31, 34	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
36	35	ad2antlr 733	. . . . . . . . . . . . . . . 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 31314	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝐵) → (-1 ·ℎ 𝑧) ∈ 𝐵)
38	10, 9, 37	mp3an12 1459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ 𝐵)
39		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘𝑣) = (normℎ‘(-1 ·ℎ 𝑧)))
40	39	oveq2d 7379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → ((normℎ‘𝑦) + (normℎ‘𝑣)) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
41		oveq2 7371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑦 +ℎ 𝑣) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
42	41	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘(𝑦 +ℎ 𝑣)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
43	42	oveq2d 7379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
44	40, 43	breq12d 5092	. . . . . . . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . 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 754	. . . . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 = (normℎ‘𝑦) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
50	49	eqcoms 2748	. . . . . . . . . . . . . . . . . 18 ⊢ ((normℎ‘𝑦) = 1 → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
51	50	ad2antll 735	. . . . . . . . . . . . . . . . 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 31112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
53	21, 11, 52	syl2an 602	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
54	53	fveq2d 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
55	54	oveq2d 7379	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
56	55	adantll 720	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
57	56	adantr 481	. . . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . . 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 11301	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℝ)
63	23	adantll 720	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
66		simpllr 781	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 0 < 𝑤)
67		lediv1 12019	. . . . . . . . . . . . . . 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 1456	. . . . . . . . . . . . . 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 842	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 11171	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
73	72	adantll 720	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
74		recn 11126	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
75	74	ad3antrrr 736	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℂ)
76	1	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ≠ 0)
77	73, 75, 76	divcan3d 11934	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤) = (normℎ‘(𝑦 −ℎ 𝑧)))
78	77	adantr 481	. . . . . . . . . . 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 5105	. . . . . . . . . 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 437	. . . . . . . . 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 3138	. . . . . . 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 3152	. . . . . 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 455	. . . . 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 520	. . . 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 413	. . 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 5083	. . . . 5 ⊢ (𝑥 = (1 / 𝑤) → (0 < 𝑥 ↔ 0 < (1 / 𝑤)))
88		breq1 5082	. . . . . . 7 ⊢ (𝑥 = (1 / 𝑤) → (𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)) ↔ (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))
89	88	imbi2d 341	. . . . . 6 ⊢ (𝑥 = (1 / 𝑤) → (((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))) ↔ ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
90	89	2ralbidv 3204	. . . . 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 638	. . . 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 3567	. . 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 3134	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 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064   class class class wbr 5079  ‘cfv 6492  (class class class)co 7363  ℂcc 11034  ℝcr 11035  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041   < clt 11177   ≤ cle 11178  -cneg 11376   / cdiv 11805   ℋchba 31015   +_ℎ cva 31016   ·_ℎ csm 31017  norm_ℎcno 31019   −_ℎ cmv 31021   S_ℋ csh 31024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114  ax-hilex 31095  ax-hfvadd 31096  ax-hv0cl 31099  ax-hfvmul 31101  ax-hvmul0 31106  ax-hfi 31175  ax-his1 31178  ax-his3 31180  ax-his4 31181

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9352  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-rp 12941  df-seq 13962  df-exp 14022  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-hnorm 31064  df-hvsub 31067  df-sh 31303

This theorem is referenced by: (None)