Theorem cdj1i 32582

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 11649	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 𝑤 ≠ 0)
2		rereccl 11906	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℝ)
3	1, 2	syldan 600	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → (1 / 𝑤) ∈ ℝ)
4	3	adantrr 727	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → (1 / 𝑤) ∈ ℝ)
5		recgt0 12034	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 0 < (1 / 𝑤))
6	5	adantrr 727	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → 0 < (1 / 𝑤))
7		1red 11179	. . . . . . . . . . . . . . . 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 11178	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
9		neg1cn 12177	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℂ
10		cdj1.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ Sℋ
11	10	sheli 31363	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ℋ)
12		hvmulcl 31162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℋ) → (-1 ·ℎ 𝑧) ∈ ℋ)
13	9, 11, 12	sylancr 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ ℋ)
14		normcl 31274	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ)
17		readdcl 11153	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
18	8, 16, 17	sylancr 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
19	18	adantr 484	. . . . . . . . . . . . . . . 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 31363	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
22		hvsubcl 31166	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) ∈ ℋ)
23	21, 11, 22	syl2an 605	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) ∈ ℋ)
24		normcl 31274	. . . . . . . . . . . . . . . . . . . 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 11155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
27	25, 26	sylan2 602	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
28	27	anassrs 471	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
29	28	adantr 484	. . . . . . . . . . . . . . . 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 31275	. . . . . . . . . . . . . . . . . . 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 11694	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
33	8, 32	mpan 700	. . . . . . . . . . . . . . . . . . 19 ⊢ ((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
34	33	biimpa 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ ∧ 0 ≤ (normℎ‘(-1 ·ℎ 𝑧))) → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
35	15, 31, 34	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
36	35	ad2antlr 737	. . . . . . . . . . . . . . . 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 31367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝐵) → (-1 ·ℎ 𝑧) ∈ 𝐵)
38	10, 9, 37	mp3an12 1471	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ 𝐵)
39		fveq2 6863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘𝑣) = (normℎ‘(-1 ·ℎ 𝑧)))
40	39	oveq2d 7408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → ((normℎ‘𝑦) + (normℎ‘𝑣)) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
41		oveq2 7400	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑦 +ℎ 𝑣) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
42	41	fveq2d 6867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘(𝑦 +ℎ 𝑣)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
43	42	oveq2d 7408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
44	40, 43	breq12d 5112	. . . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . . . 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 7399	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 = (normℎ‘𝑦) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
50	49	eqcoms 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((normℎ‘𝑦) = 1 → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
51	50	ad2antll 739	. . . . . . . . . . . . . . . . 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 31165	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
53	21, 11, 52	syl2an 605	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
54	53	fveq2d 6867	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
55	54	oveq2d 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
56	55	adantll 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
57	56	adantr 484	. . . . . . . . . . . . . . . . 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 5131	. . . . . . . . . . . . . . . 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 11337	. . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℝ)
63	23	adantll 724	. . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
66		simpllr 785	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 0 < 𝑤)
67		lediv1 12054	. . . . . . . . . . . . . . 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 1468	. . . . . . . . . . . . . 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 847	. . . . . . . . . . . . 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 241	. . . . . . . . . . . 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 410	. . . . . . . . . . 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 11207	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
73	72	adantll 724	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
74		recn 11160	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
75	74	ad3antrrr 740	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℂ)
76	1	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ≠ 0)
77	73, 75, 76	divcan3d 11969	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → ((𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) / 𝑤) = (normℎ‘(𝑦 −ℎ 𝑧)))
78	77	adantr 484	. . . . . . . . . . 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 5125	. . . . . . . . . 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 440	. . . . . . . . 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 3159	. . . . . . 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 3173	. . . . . 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 458	. . . . 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 523	. . . 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 416	. . 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 5103	. . . . 5 ⊢ (𝑥 = (1 / 𝑤) → (0 < 𝑥 ↔ 0 < (1 / 𝑤)))
88		breq1 5102	. . . . . . 7 ⊢ (𝑥 = (1 / 𝑤) → (𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)) ↔ (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))
89	88	imbi2d 342	. . . . . 6 ⊢ (𝑥 = (1 / 𝑤) → (((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))) ↔ ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
90	89	2ralbidv 3225	. . . . 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 641	. . . 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 3581	. . 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 3155	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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  ℂcc 11068  ℝcr 11069  0cc0 11070  1c1 11071   + caddc 11073   · cmul 11075   < clt 11213   ≤ cle 11214  -cneg 11412   / cdiv 11841   ℋchba 31068   +_ℎ cva 31069   ·_ℎ csm 31070  norm_ℎcno 31072   −_ℎ cmv 31074   S_ℋ csh 31077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148  ax-hilex 31148  ax-hfvadd 31149  ax-hv0cl 31152  ax-hfvmul 31154  ax-hvmul0 31159  ax-hfi 31228  ax-his1 31231  ax-his3 31233  ax-his4 31234

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-sup 9385  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-seq 14012  df-exp 14072  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-hnorm 31117  df-hvsub 31120  df-sh 31356

This theorem is referenced by: (None)