Theorem cdj1i 32362

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 11643	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 𝑤 ≠ 0)
2		rereccl 11900	. . . . . . 7 ⊢ ((𝑤 ∈ ℝ ∧ 𝑤 ≠ 0) → (1 / 𝑤) ∈ ℝ)
3	1, 2	syldan 591	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → (1 / 𝑤) ∈ ℝ)
4	3	adantrr 717	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → (1 / 𝑤) ∈ ℝ)
5		recgt0 12028	. . . . . 6 ⊢ ((𝑤 ∈ ℝ ∧ 0 < 𝑤) → 0 < (1 / 𝑤))
6	5	adantrr 717	. . . . 5 ⊢ ((𝑤 ∈ ℝ ∧ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))))) → 0 < (1 / 𝑤))
7		1red 11175	. . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
9		neg1cn 12171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -1 ∈ ℂ
10		cdj1.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐵 ∈ Sℋ
11	10	sheli 31143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ 𝐵 → 𝑧 ∈ ℋ)
12		hvmulcl 30942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((-1 ∈ ℂ ∧ 𝑧 ∈ ℋ) → (-1 ·ℎ 𝑧) ∈ ℋ)
13	9, 11, 12	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ ℋ)
14		normcl 31054	. . . . . . . . . . . . . . . . . . . 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 11151	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) ∈ ℝ)
18	8, 16, 17	sylancr 587	. . . . . . . . . . . . . . . . 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 31143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
22		hvsubcl 30946	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) ∈ ℋ)
23	21, 11, 22	syl2an 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) ∈ ℋ)
24		normcl 31054	. . . . . . . . . . . . . . . . . . . 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 11153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ ℝ ∧ (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℝ) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
27	25, 26	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℝ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
28	27	anassrs 467	. . . . . . . . . . . . . . . . 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 31055	. . . . . . . . . . . . . . . . . . 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 11688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ (normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ) → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
33	8, 32	mpan 690	. . . . . . . . . . . . . . . . . . 19 ⊢ ((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ → (0 ≤ (normℎ‘(-1 ·ℎ 𝑧)) ↔ 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧)))))
34	33	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((normℎ‘(-1 ·ℎ 𝑧)) ∈ ℝ ∧ 0 ≤ (normℎ‘(-1 ·ℎ 𝑧))) → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
35	15, 31, 34	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝐵 → 1 ≤ (1 + (normℎ‘(-1 ·ℎ 𝑧))))
36	35	ad2antlr 727	. . . . . . . . . . . . . . . 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 31147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ Sℋ ∧ -1 ∈ ℂ ∧ 𝑧 ∈ 𝐵) → (-1 ·ℎ 𝑧) ∈ 𝐵)
38	10, 9, 37	mp3an12 1453	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ 𝐵 → (-1 ·ℎ 𝑧) ∈ 𝐵)
39		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘𝑣) = (normℎ‘(-1 ·ℎ 𝑧)))
40	39	oveq2d 7403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → ((normℎ‘𝑦) + (normℎ‘𝑣)) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
41		oveq2 7395	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑦 +ℎ 𝑣) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
42	41	fveq2d 6862	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (normℎ‘(𝑦 +ℎ 𝑣)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
43	42	oveq2d 7403	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = (-1 ·ℎ 𝑧) → (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
44	40, 43	breq12d 5120	. . . . . . . . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . 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 748	. . . . . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 = (normℎ‘𝑦) → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
50	49	eqcoms 2737	. . . . . . . . . . . . . . . . . 18 ⊢ ((normℎ‘𝑦) = 1 → (1 + (normℎ‘(-1 ·ℎ 𝑧))) = ((normℎ‘𝑦) + (normℎ‘(-1 ·ℎ 𝑧))))
51	50	ad2antll 729	. . . . . . . . . . . . . . . . 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 30945	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
53	21, 11, 52	syl2an 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑦 −ℎ 𝑧) = (𝑦 +ℎ (-1 ·ℎ 𝑧)))
54	53	fveq2d 6862	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) = (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧))))
55	54	oveq2d 7403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) = (𝑤 · (normℎ‘(𝑦 +ℎ (-1 ·ℎ 𝑧)))))
56	55	adantll 714	. . . . . . . . . . . . . . . . . 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 5139	. . . . . . . . . . . . . . . 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 11331	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 719	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℝ)
63	23	adantll 714	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (𝑤 · (normℎ‘(𝑦 −ℎ 𝑧))) ∈ ℝ)
66		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 0 < 𝑤)
67		lediv1 12048	. . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 11202	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
73	72	adantll 714	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → (normℎ‘(𝑦 −ℎ 𝑧)) ∈ ℂ)
74		recn 11158	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
75	74	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ∈ ℂ)
76	1	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ ℝ ∧ 0 < 𝑤) ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐵) → 𝑤 ≠ 0)
77	73, 75, 76	divcan3d 11963	. . . . . . . . . . . 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 5133	. . . . . . . . . 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 436	. . . . . . . . 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 3131	. . . . . . 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 3145	. . . . . 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 454	. . . . 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 515	. . . 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 412	. . 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 5111	. . . . 5 ⊢ (𝑥 = (1 / 𝑤) → (0 < 𝑥 ↔ 0 < (1 / 𝑤)))
88		breq1 5110	. . . . . . 7 ⊢ (𝑥 = (1 / 𝑤) → (𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)) ↔ (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧))))
89	88	imbi2d 340	. . . . . 6 ⊢ (𝑥 = (1 / 𝑤) → (((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧))) ↔ ((normℎ‘𝑦) = 1 → (1 / 𝑤) ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
90	89	2ralbidv 3201	. . . . 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 632	. . . 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 3588	. . 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 3127	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073   < clt 11208   ≤ cle 11209  -cneg 11406   / cdiv 11835   ℋchba 30848   +_ℎ cva 30849   ·_ℎ csm 30850  norm_ℎcno 30852   −_ℎ cmv 30854   S_ℋ csh 30857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-hilex 30928  ax-hfvadd 30929  ax-hv0cl 30932  ax-hfvmul 30934  ax-hvmul0 30939  ax-hfi 31008  ax-his1 31011  ax-his3 31013  ax-his4 31014

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-sup 9393  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-seq 13967  df-exp 14027  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-hnorm 30897  df-hvsub 30900  df-sh 31136

This theorem is referenced by: (None)