Theorem cdj3lem2b 31677

Description: Lemma for cdj3i 31681. 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 31674	. 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 30556	. . . . . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
7	6	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
8		fvoveq1 7428	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
9	8	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
10	7, 9	breq12d 5160	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
11		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
12	11	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
13		oveq2 7413	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
14	13	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
15	14	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
16	12, 15	breq12d 5160	. . . . . . . . . . . 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 3621	. . . . . . . . . . 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 31675	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
20	19	3expa 1118	. . . . . . . . . . . . . . . 16 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
21	20	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
22	21	ad2ant2r 745	. . . . . . . . . . . . . 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 30454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
24		normge0 30366	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → 0 ≤ (normℎ‘ℎ))
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → 0 ≤ (normℎ‘ℎ))
26	25	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → 0 ≤ (normℎ‘ℎ))
27	1	sheli 30454	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
28		normcl 30365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
30		normcl 30365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
31	23, 30	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
32		addge01 11720	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → (0 ≤ (normℎ‘ℎ) ↔ (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ))))
33	29, 31, 32	syl2an 596	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ≤ ((normℎ‘𝑡) + (normℎ‘ℎ)))
36	29	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (normℎ‘𝑡) ∈ ℝ)
37		readdcl 11189	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
38	29, 31, 37	syl2an 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
39	38	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ∈ ℝ)
40		hvaddcl 30252	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈ ℋ)
41	27, 23, 40	syl2an 596	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ)
42		normcl 30365	. . . . . . . . . . . . . . . . . . . . . 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 11191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
45	43, 44	sylan2 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
46	45	ancoms 459	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ 𝑣 ∈ ℝ) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
47		letr 11304	. . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 650	. . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . 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 5169	. . . . . . . . . . . . 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 6893	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
55		fveq2 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
56	55	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑣 · (normℎ‘𝑢)) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
57	54, 56	breq12d 5160	. . . . . . . . . . . . 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 420	. . . . . . . . . . 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 3210	. . . . . . 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 3152	. . . 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 612	. . 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 3168	. 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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ∩ cin 3946   class class class wbr 5147   ↦ cmpt 5230  ‘cfv 6540  ℩crio 7360  (class class class)co 7405  ℝcr 11105  0cc0 11106   + caddc 11109   · cmul 11111   < clt 11244   ≤ cle 11245   ℋchba 30159   +_ℎ cva 30160  norm_ℎcno 30163   S_ℋ csh 30168   +_ℋ cph 30171  0_ℋc0h 30175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-hilex 30239  ax-hfvadd 30240  ax-hvcom 30241  ax-hvass 30242  ax-hv0cl 30243  ax-hvaddid 30244  ax-hfvmul 30245  ax-hvmulid 30246  ax-hvmulass 30247  ax-hvdistr1 30248  ax-hvdistr2 30249  ax-hvmul0 30250  ax-hfi 30319  ax-his1 30322  ax-his3 30324  ax-his4 30325

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-grpo 29733  df-ablo 29785  df-hnorm 30208  df-hvsub 30211  df-sh 30447  df-ch0 30493  df-shs 30548

This theorem is referenced by:  cdj3lem3b  31680  cdj3i  31681