Theorem cdj3lem2b 32124

Description: Lemma for cdj3i 32128. 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 32121	. 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 31003	. . . . . . . 8 ⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) ↔ ∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ))
5	4	biimpi 215	. . . . . . 7 ⊢ (𝑢 ∈ (𝐴 +ℋ 𝐵) → ∃𝑡 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑢 = (𝑡 +ℎ ℎ))
6		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
7	6	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
8		fvoveq1 7435	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
9	8	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
10	7, 9	breq12d 5161	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
11		fveq2 6891	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
12	11	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
13		oveq2 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
14	13	fveq2d 6895	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
15	14	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
16	12, 15	breq12d 5161	. . . . . . . . . . . 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 3622	. . . . . . . . . . 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 32122	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
20	19	3expa 1117	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
21	20	fveq2d 6895	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
22	21	ad2ant2r 744	. . . . . . . . . . . . . 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 30901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
24		normge0 30813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → 0 ≤ (normℎ‘ℎ))
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → 0 ≤ (normℎ‘ℎ))
26	25	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → 0 ≤ (normℎ‘ℎ))
27	1	sheli 30901	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
28		normcl 30812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
30		normcl 30812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
31	23, 30	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
32		addge01 11731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → (0 ≤ (normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ))))
33	29, 31, 32	syl2an 595	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (0 ≤ (normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ))))
34	26, 33	mpbid 231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)))
35	34	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)))
36	29	ad2antrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ∈ ℝ)
37		readdcl 11199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
38	29, 31, 37	syl2an 595	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
39	38	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
40		hvaddcl 30699	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈ ℋ)
41	27, 23, 40	syl2an 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ)
42		normcl 30812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 +ℎ ℎ) ∈ ℋ → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
44		remulcl 11201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
45	43, 44	sylan2 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
46	45	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
47		letr 11315	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 649	. . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . 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 5170	. . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
55		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
56	55	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑣 · (normℎ‘𝑢)) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
57	54, 56	breq12d 5161	. . . . . . . . . . . . 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 246	. . . . . . . . . . . 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 419	. . . . . . . . . . 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 47	. . . . . . . . . 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 96	. . . . . . . . 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 93	. . . . . . . 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 3209	. . . . . . 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 31	. . . . . 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 89	. . . . 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 3151	. . . 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 611	. . 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 3167	. 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 38	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 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   ∩ cin 3947   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6543  ℩crio 7367  (class class class)co 7412  ℝcr 11115  0cc0 11116   + caddc 11119   · cmul 11121   < clt 11255   ≤ cle 11256   ℋchba 30606   +_ℎ cva 30607  norm_ℎcno 30610   S_ℋ csh 30615   +_ℋ cph 30618  0_ℋc0h 30622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-hilex 30686  ax-hfvadd 30687  ax-hvcom 30688  ax-hvass 30689  ax-hv0cl 30690  ax-hvaddid 30691  ax-hfvmul 30692  ax-hvmulid 30693  ax-hvmulass 30694  ax-hvdistr1 30695  ax-hvdistr2 30696  ax-hvmul0 30697  ax-hfi 30766  ax-his1 30769  ax-his3 30771  ax-his4 30772

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-sup 9443  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-rp 12982  df-seq 13974  df-exp 14035  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-grpo 30180  df-ablo 30232  df-hnorm 30655  df-hvsub 30658  df-sh 30894  df-ch0 30940  df-shs 30995

This theorem is referenced by:  cdj3lem3b  32127  cdj3i  32128