Theorem cdj3lem2b 33096

Description: Lemma for cdj3i 33100. The first-component function 𝑆 is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdj3lem2.1	⊢ 𝐴 ∈ Sℋ
cdj3lem2.2	⊢ 𝐵 ∈ Sℋ
cdj3lem2.3	⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))

Assertion

Ref	Expression
cdj3lem2b	⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))

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

Allowed substitution hints:   𝑆(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem cdj3lem2b

Dummy variables 𝑡 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdj3lem2.1	. . 3 ⊢ 𝐴 ∈ Sℋ
2		cdj3lem2.2	. . 3 ⊢ 𝐵 ∈ Sℋ
3	1, 2	cdj3lem1 33093	. 2 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → (𝐴 ∩ 𝐵) = 0ℋ)
4	1, 2	shseli 31975	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ))
5	4	biimpi 219	. . . . . . 7 ⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) → ∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ))
6		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
7	6	oveq1d 7466	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
8		fvoveq1 7474	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
9	8	oveq2d 7467	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
10	7, 9	breq12d 5121	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
11		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
12	11	oveq2d 7467	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
13		oveq2 7459	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
14	13	fveq2d 6920	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
15	14	oveq2d 7467	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
16	12, 15	breq12d 5121	. . . . . . . . . . . 12 ⊢ (𝑦 = ℎ → (((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
17	10, 16	rspc2v 3590	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
18		cdj3lem2.3	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))
19	1, 2, 18	cdj3lem2 33094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
20	19	3expa 1136	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
21	20	fveq2d 6920	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
22	21	ad2ant2r 760	. . . . . . . . . . . . . 14 ⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
23	2	sheli 31873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
24		normge0 31785	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → 0 ≤ (normℎ‘ℎ))
25	23, 24	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → 0 ≤ (normℎ‘ℎ))
26	25	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → 0 ≤ (normℎ‘ℎ))
27	1	sheli 31873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
28		normcl 31784	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
29	27, 28	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
30		normcl 31784	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
31	23, 30	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
32		addge01 11866	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → (0 ≤ (normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ))))
33	29, 31, 32	syl2an 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (0 ≤ (normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ))))
34	26, 33	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)))
35	34	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)))
36	29	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ∈ ℝ)
37		readdcl 11322	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
38	29, 31, 37	syl2an 608	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
39	38	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
40		hvaddcl 31671	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈ ℋ)
41	27, 23, 40	syl2an 608	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ)
42		normcl 31784	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 +ℎ ℎ) ∈ ℋ → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
43	41, 42	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
44		remulcl 11324	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
45	43, 44	sylan2 605	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
46	45	ancoms 464	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
47		letr 11443	. . . . . . . . . . . . . . . . . . 19 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ ∧ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ) → (((normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) → (normℎ‘𝑡) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
48	36, 39, 46, 47	syl3anc 1399	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (((normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) → (normℎ‘𝑡) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
49	35, 48	mpand 708	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) → (normℎ‘𝑡) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
50	49	imp 412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) → (normℎ‘𝑡) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
51	50	an32s 665	. . . . . . . . . . . . . . 15 ⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
52	51	adantrl 729	. . . . . . . . . . . . . 14 ⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) → (normℎ‘𝑡) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
53	22, 52	eqbrtrd 5133	. . . . . . . . . . . . 13 ⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
54		2fveq3 6921	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
55		fveq2 6916	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
56	55	oveq2d 7467	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑣 · (normℎ‘𝑢)) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
57	54, 56	breq12d 5121	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → ((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
58	53, 57	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))) ∧ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ)) → (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
59	58	exp31 425	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))))
60	17, 59	syld 48	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))))
61	60	com14 97	. . . . . . . . 9 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))))
62	61	com4t 94	. . . . . . . 8 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑢 = (𝑡 +ℎ ℎ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))))
63	62	rexlimdvv 3222	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))))
64	5, 63	syl5com 32	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) → (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))))
65	64	com3l 90	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → (𝑢 ∈ (𝐴 +ℋ 𝐵) → (normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))))
66	65	ralrimdv 3164	. . . 4 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) → ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
67	66	anim2d 624	. . 3 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝑣 ∈ ℝ) → ((0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))))
68	67	reximdva 3179	. 2 ⊢ ((𝐴 ∩ 𝐵) = 0ℋ → (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)))))
69	3, 68	mpcom 39	1 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∩ cin 4092   class class class wbr 5108   ↦ cmpt 5192  ‘cfv 6566  ℩crio 7407  (class class class)co 7451  ℝcr 11238  0cc0 11239   + caddc 11242   · cmul 11244   < clt 11382   ≤ cle 11383   ℋchba 31578   +_ℎ cva 31579  norm_ℎcno 31582   S_ℋ csh 31587   +_ℋ cph 31590  0_ℋc0h 31594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317  ax-hilex 31658  ax-hfvadd 31659  ax-hvcom 31660  ax-hvass 31661  ax-hv0cl 31662  ax-hvaddid 31663  ax-hfvmul 31664  ax-hvmulid 31665  ax-hvmulass 31666  ax-hvdistr1 31667  ax-hvdistr2 31668  ax-hvmul0 31669  ax-hfi 31738  ax-his1 31741  ax-his3 31743  ax-his4 31744

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-rp 13161  df-seq 14185  df-exp 14245  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-grpo 31152  df-ablo 31204  df-hnorm 31627  df-hvsub 31630  df-sh 31866  df-ch0 31912  df-shs 31967

This theorem is used by:  cdj3lem3b  33099  cdj3i  33100