Theorem cdj3i 32925

Description: Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdj3.1	⊢ 𝐴 ∈ Sℋ
cdj3.2	⊢ 𝐵 ∈ Sℋ
cdj3.3	⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))
cdj3.4	⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
cdj3.5	⊢ (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
cdj3.6	⊢ (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))

Assertion

Ref	Expression
cdj3i	⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢)   𝜓(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢)   𝑆(𝑥, 𝑦, 𝑧, 𝑤)   𝑇(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem cdj3i

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

Step	Hyp	Ref	Expression
1		cdj3.1	. . . 4 ⊢ 𝐴 ∈ Sℋ
2		cdj3.2	. . . 4 ⊢ 𝐵 ∈ Sℋ
3	1, 2	cdj3lem1 32918	. . 3 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → (𝐴 ∩ 𝐵) = 0ℋ)
4		cdj3.3	. . . . 5 ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))
5	1, 2, 4	cdj3lem2b 32921	. . . 4 ⊢ (∃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))))
6		cdj3.5	. . . 4 ⊢ (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
7	5, 6	sylibr 237	. . 3 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜑)
8		cdj3.4	. . . . 5 ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))
9	1, 2, 8	cdj3lem3b 32924	. . . 4 ⊢ (∃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‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
10		cdj3.6	. . . 4 ⊢ (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
11	9, 10	sylibr 237	. . 3 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜓)
12	3, 7, 11	3jca 1146	. 2 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓))
13		breq2 5107	. . . . . . . . 9 ⊢ (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14		oveq1 7421	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (𝑣 · (normℎ‘𝑢)) = (𝑓 · (normℎ‘𝑢)))
15	14	breq2d 5115	. . . . . . . . . 10 ⊢ (𝑣 = 𝑓 → ((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
16	15	ralbidv 3185	. . . . . . . . 9 ⊢ (𝑣 = 𝑓 → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
17	13, 16	anbi12d 644	. . . . . . . 8 ⊢ (𝑣 = 𝑓 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))) ↔ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)))))
18	17	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))) ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
19	6, 18	bitri 278	. . . . . 6 ⊢ (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
20		breq2 5107	. . . . . . . . 9 ⊢ (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21		oveq1 7421	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑔 → (𝑣 · (normℎ‘𝑢)) = (𝑔 · (normℎ‘𝑢)))
22	21	breq2d 5115	. . . . . . . . . 10 ⊢ (𝑣 = 𝑔 → ((normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
23	22	ralbidv 3185	. . . . . . . . 9 ⊢ (𝑣 = 𝑔 → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
24	20, 23	anbi12d 644	. . . . . . . 8 ⊢ (𝑣 = 𝑔 → ((0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))) ↔ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))))
25	24	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))) ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
26	10, 25	bitri 278	. . . . . 6 ⊢ (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
27	19, 26	anbi12i 640	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) ↔ (∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))))
28		reeanv 3234	. . . . 5 ⊢ (∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))) ↔ (∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))))
29	27, 28	bitr4i 281	. . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ ∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))))
30		an4 669	. . . . . 6 ⊢ (((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))) ↔ ((0 < 𝑓 ∧ 0 < 𝑔) ∧ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))))
31		addgt0 11727	. . . . . . . . 9 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 < 𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔))
32	31	ex 418	. . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
33	32	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
34	1, 2	shsvai 31848	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵))
35		2fveq3 6884	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
36		fveq2 6879	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
37	36	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑓 · (normℎ‘𝑢)) = (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))))
38	35, 37	breq12d 5116	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → ((normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ)))))
39	38	rspcv 3572	. . . . . . . . . . . 12 ⊢ ((𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵) → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ)))))
40		2fveq3 6884	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑇‘𝑢)) = (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))))
41	36	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑔 · (normℎ‘𝑢)) = (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))
42	40, 41	breq12d 5116	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → ((normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
43	42	rspcv 3572	. . . . . . . . . . . 12 ⊢ ((𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵) → (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)) → (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
44	39, 43	anim12d 621	. . . . . . . . . . 11 ⊢ ((𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))) → ((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
45	34, 44	syl 18	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))) → ((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
46	45	adantl 487	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))) → ((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
47	1	sheli 31698	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
48		normcl 31609	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
49	47, 48	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
50	2	sheli 31698	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
51		normcl 31609	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
52	50, 51	syl 18	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
53	49, 52	anim12i 625	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ))
54	53	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ))
55		hvaddcl 31496	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈ ℋ)
56	47, 50, 55	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ)
57		normcl 31609	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 +ℎ ℎ) ∈ ℋ → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
58	56, 57	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
59		remulcl 11212	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
60	58, 59	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
61	60	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
62		remulcl 11212	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
63	58, 62	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
64	63	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
65		le2add 11723	. . . . . . . . . . . 12 ⊢ ((((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ) ∧ ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ ∧ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)) → (((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
66	54, 61, 64, 65	syl12anc 850	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
67	66	adantll 727	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
68	1, 2, 4	cdj3lem2 32919	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
69	68	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
70	69	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ↔ (normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ)))))
71	1, 2, 8	cdj3lem3 32922	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝑡 +ℎ ℎ)) = ℎ)
72	71	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) = (normℎ‘ℎ))
73	72	breq1d 5113	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ((normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ↔ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
74	70, 73	anbi12d 644	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) ↔ ((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
75	74	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) ↔ ((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
76	75	ancoms 464	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) ↔ ((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
77	76	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) ↔ ((normℎ‘𝑡) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘ℎ) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
78		recn 11217	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
79		recn 11217	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
80	58	recnd 11264	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ)
81		adddir 11224	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℂ ∧ 𝑔 ∈ ℂ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ) → ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
82	78, 79, 80, 81	syl3an 1178	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
83	82	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
84	83	breq2d 5115	. . . . . . . . . . 11 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
85	84	adantll 727	. . . . . . . . . 10 ⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) + (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))))
86	67, 77, 85	3imtr4d 297	. . . . . . . . 9 ⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (((normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∧ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))))
87	46, 86	syld 48	. . . . . . . 8 ⊢ ((((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))) → ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))))
88	87	ralrimdvva 3217	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))) → ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))))
89		readdcl 11210	. . . . . . . . 9 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
90		breq2 5107	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
91		fveq2 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
92	91	oveq1d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
93		fvoveq1 7437	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
94	93	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
95	92, 94	breq12d 5116	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
96		fveq2 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
97	96	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
98		oveq2 7422	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
99	98	fveq2d 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
100	99	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
101	97, 100	breq12d 5116	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ℎ → (((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ)))))
102	95, 101	cbvral2vw 3244	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
103		oveq1 7421	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑓 + 𝑔) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))))
104	103	breq2d 5115	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑓 + 𝑔) → (((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ↔ ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))))
105	104	2ralbidv 3226	. . . . . . . . . . . . 13 ⊢ (𝑣 = (𝑓 + 𝑔) → (∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))))
106	102, 105	bitrid 286	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑓 + 𝑔) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))))
107	90, 106	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑣 = (𝑓 + 𝑔) → ((0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ (0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))))))
108	107	rspcev 3576	. . . . . . . . . 10 ⊢ (((𝑓 + 𝑔) ∈ ℝ ∧ (0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))))
109	108	ex 418	. . . . . . . . 9 ⊢ ((𝑓 + 𝑔) ∈ ℝ → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
110	89, 109	syl 18	. . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
111	110	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0 < (𝑓 + 𝑔) ∧ ∀𝑡 ∈ 𝐴 ∀ℎ ∈ 𝐵 ((normℎ‘𝑡) + (normℎ‘ℎ)) ≤ ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
112	33, 88, 111	syl2and 620	. . . . . 6 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (((0 < 𝑓 ∧ 0 < 𝑔) ∧ (∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
113	30, 112	biimtrid 245	. . . . 5 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
114	113	rexlimdvva 3219	. . . 4 ⊢ ((𝐴 ∩ 𝐵) = 0ℋ → (∃𝑓 ∈ ℝ ∃𝑔 ∈ ℝ ((0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))) ∧ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
115	29, 114	biimtrid 245	. . 3 ⊢ ((𝐴 ∩ 𝐵) = 0ℋ → ((𝜑 ∧ 𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))))))
116	115	3impib 1134	. 2 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))))
117	12, 116	impbii 212	1 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6533  ℩crio 7370  (class class class)co 7414  ℂcc 11125  ℝcr 11126  0cc0 11127   + caddc 11130   · cmul 11132   < clt 11270   ≤ cle 11271   ℋchba 31403   +_ℎ cva 31404  norm_ℎcno 31407   S_ℋ csh 31412   +_ℋ cph 31415  0_ℋc0h 31419

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 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205  ax-hilex 31483  ax-hfvadd 31484  ax-hvcom 31485  ax-hvass 31486  ax-hv0cl 31487  ax-hvaddid 31488  ax-hfvmul 31489  ax-hvmulid 31490  ax-hvmulass 31491  ax-hvdistr1 31492  ax-hvdistr2 31493  ax-hvmul0 31494  ax-hfi 31563  ax-his1 31566  ax-his3 31568  ax-his4 31569

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 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-rp 13046  df-seq 14069  df-exp 14129  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-grpo 30977  df-ablo 31029  df-hnorm 31452  df-hvsub 31455  df-sh 31691  df-ch0 31737  df-shs 31792

This theorem is used by: (None)