Theorem cdj3lem2b 32918

Description: Lemma for cdj3i 32922. 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 32915	. 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 31797	. . . . . . . 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 6878	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
7	6	oveq1d 7428	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
8		fvoveq1 7436	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
9	8	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
10	7, 9	breq12d 5116	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
11		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
12	11	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
13		oveq2 7421	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
14	13	fveq2d 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
15	14	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
16	12, 15	breq12d 5116	. . . . . . . . . . . 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 3587	. . . . . . . . . . 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 32916	. . . . . . . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 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 31695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
24		normge0 31607	. . . . . . . . . . . . . . . . . . . . . 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 31695	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
28		normcl 31606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
29	27, 28	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
30		normcl 31606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
31	23, 30	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
32		addge01 11748	. . . . . . . . . . . . . . . . . . . . 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 11207	. . . . . . . . . . . . . . . . . . . . 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 31493	. . . . . . . . . . . . . . . . . . . . . . 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 31606	. . . . . . . . . . . . . . . . . . . . . 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 11209	. . . . . . . . . . . . . . . . . . . . 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 11328	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
55		fveq2 6878	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
56	55	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑣 · (normℎ‘𝑢)) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
57	54, 56	breq12d 5116	. . . . . . . . . . . . 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 3218	. . . . . . 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 3160	. . . 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 3175	. 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 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  ℩crio 7369  (class class class)co 7413  ℝcr 11123  0cc0 11124   + caddc 11127   · cmul 11129   < clt 11267   ≤ cle 11268   ℋchba 31400   +_ℎ cva 31401  norm_ℎcno 31404   S_ℋ csh 31409   +_ℋ cph 31412  0_ℋc0h 31416

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-hilex 31480  ax-hfvadd 31481  ax-hvcom 31482  ax-hvass 31483  ax-hv0cl 31484  ax-hvaddid 31485  ax-hfvmul 31486  ax-hvmulid 31487  ax-hvmulass 31488  ax-hvdistr1 31489  ax-hvdistr2 31490  ax-hvmul0 31491  ax-hfi 31560  ax-his1 31563  ax-his3 31565  ax-his4 31566

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-z 12616  df-uz 12888  df-rp 13043  df-seq 14066  df-exp 14126  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-grpo 30974  df-ablo 31026  df-hnorm 31449  df-hvsub 31452  df-sh 31688  df-ch0 31734  df-shs 31789

This theorem is used by:  cdj3lem3b  32921  cdj3i  32922