Theorem cdj3lem2b 32729

Description: Lemma for cdj3i 32733. 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 32726	. 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 31608	. . . . . . . 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 6882	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
7	6	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
8		fvoveq1 7434	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
9	8	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
10	7, 9	breq12d 5126	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
11		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
12	11	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
13		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
14	13	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
15	14	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
16	12, 15	breq12d 5126	. . . . . . . . . . . 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 3601	. . . . . . . . . . 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 32727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
20	19	3expa 1134	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
21	20	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
22	21	ad2ant2r 759	. . . . . . . . . . . . . 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 31506	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
24		normge0 31418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → 0 ≤ (normℎ‘ℎ))
25	23, 24	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → 0 ≤ (normℎ‘ℎ))
26	25	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → 0 ≤ (normℎ‘ℎ))
27	1	sheli 31506	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
28		normcl 31417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
29	27, 28	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
30		normcl 31417	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
31	23, 30	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
32		addge01 11723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → (0 ≤ (normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ))))
33	29, 31, 32	syl2an 607	. . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)))
36	29	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ∈ ℝ)
37		readdcl 11182	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
38	29, 31, 37	syl2an 607	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
39	38	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
40		hvaddcl 31304	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈ ℋ)
41	27, 23, 40	syl2an 607	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ)
42		normcl 31417	. . . . . . . . . . . . . . . . . . . . . 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 11184	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
45	43, 44	sylan2 604	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
46	45	ancoms 463	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
47		letr 11303	. . . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . . 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 707	. . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 5137	. . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
55		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
56	55	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑣 · (normℎ‘𝑢)) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
57	54, 56	breq12d 5126	. . . . . . . . . . . . 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 424	. . . . . . . . . . 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 3227	. . . . . . 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 3169	. . . 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 623	. . 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 3184	. 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∩ cin 3912   class class class wbr 5113   ↦ cmpt 5196  ‘cfv 6537  ℩crio 7367  (class class class)co 7411  ℝcr 11098  0cc0 11099   + caddc 11102   · cmul 11104   < clt 11242   ≤ cle 11243   ℋchba 31211   +_ℎ cva 31212  norm_ℎcno 31215   S_ℋ csh 31220   +_ℋ cph 31223  0_ℋc0h 31227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-hilex 31291  ax-hfvadd 31292  ax-hvcom 31293  ax-hvass 31294  ax-hv0cl 31295  ax-hvaddid 31296  ax-hfvmul 31297  ax-hvmulid 31298  ax-hvmulass 31299  ax-hvdistr1 31300  ax-hvdistr2 31301  ax-hvmul0 31302  ax-hfi 31371  ax-his1 31374  ax-his3 31376  ax-his4 31377

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-seq 14037  df-exp 14097  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-grpo 30785  df-ablo 30837  df-hnorm 31260  df-hvsub 31263  df-sh 31499  df-ch0 31545  df-shs 31600

This theorem is referenced by:  cdj3lem3b  32732  cdj3i  32733