Theorem cdj3i 32465

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 32458	. . 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 32461	. . . 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 234	. . 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 32464	. . . 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 234	. . 3 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → 𝜓)
12	3, 7, 11	3jca 1128	. 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 5100	. . . . . . . . 9 ⊢ (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (𝑣 · (normℎ‘𝑢)) = (𝑓 · (normℎ‘𝑢)))
15	14	breq2d 5108	. . . . . . . . . 10 ⊢ (𝑣 = 𝑓 → ((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
16	15	ralbidv 3157	. . . . . . . . 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 632	. . . . . . . 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 3213	. . . . . . 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 275	. . . . . 6 ⊢ (𝜑 ↔ ∃𝑓 ∈ ℝ (0 < 𝑓 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
20		breq2 5100	. . . . . . . . 9 ⊢ (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑔 → (𝑣 · (normℎ‘𝑢)) = (𝑔 · (normℎ‘𝑢)))
22	21	breq2d 5108	. . . . . . . . . 10 ⊢ (𝑣 = 𝑔 → ((normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
23	22	ralbidv 3157	. . . . . . . . 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 632	. . . . . . . 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 3213	. . . . . . 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 275	. . . . . 6 ⊢ (𝜓 ↔ ∃𝑔 ∈ ℝ (0 < 𝑔 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
27	19, 26	anbi12i 628	. . . . 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 3206	. . . . 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 278	. . . 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 656	. . . . . 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 11621	. . . . . . . . 9 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (0 < 𝑓 ∧ 0 < 𝑔)) → 0 < (𝑓 + 𝑔))
32	31	ex 412	. . . . . . . 8 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
33	32	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) = 0ℋ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → ((0 < 𝑓 ∧ 0 < 𝑔) → 0 < (𝑓 + 𝑔)))
34	1, 2	shsvai 31388	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵))
35		2fveq3 6837	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
36		fveq2 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
37	36	oveq2d 7372	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑓 · (normℎ‘𝑢)) = (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))))
38	35, 37	breq12d 5109	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → ((normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) ≤ (𝑓 · (normℎ‘(𝑡 +ℎ ℎ)))))
39	38	rspcv 3570	. . . . . . . . . . . 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 6837	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑇‘𝑢)) = (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))))
41	36	oveq2d 7372	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑔 · (normℎ‘𝑢)) = (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))
42	40, 41	breq12d 5109	. . . . . . . . . . . . 13 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → ((normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) ≤ (𝑔 · (normℎ‘(𝑡 +ℎ ℎ)))))
43	42	rspcv 3570	. . . . . . . . . . . 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 609	. . . . . . . . . . 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 17	. . . . . . . . . 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 481	. . . . . . . . 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 31238	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
48		normcl 31149	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
50	2	sheli 31238	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
51		normcl 31149	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ ℋ → (normℎ‘ℎ) ∈ ℝ)
52	50, 51	syl 17	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ 𝐵 → (normℎ‘ℎ) ∈ ℝ)
53	49, 52	anim12i 613	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ))
54	53	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ((normℎ‘𝑡) ∈ ℝ ∧ (normℎ‘ℎ) ∈ ℝ))
55		hvaddcl 31036	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ ℋ ∧ ℎ ∈ ℋ) → (𝑡 +ℎ ℎ) ∈ ℋ)
56	47, 50, 55	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ ℋ)
57		normcl 31149	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 +ℎ ℎ) ∈ ℋ → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
58	56, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ)
59		remulcl 11109	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
60	58, 59	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
61	60	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
62		remulcl 11109	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∈ ℝ ∧ (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℝ) → (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
63	58, 62	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ ℝ ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
64	63	adantll 714	. . . . . . . . . . . 12 ⊢ (((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) ∧ (𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))) ∈ ℝ)
65		le2add 11617	. . . . . . . . . . . 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 836	. . . . . . . . . . 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 714	. . . . . . . . . 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 32459	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
69	68	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
70	69	breq1d 5106	. . . . . . . . . . . . . 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 32462	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝑡 +ℎ ℎ)) = ℎ)
72	71	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) = (normℎ‘ℎ))
73	72	breq1d 5106	. . . . . . . . . . . . . 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 632	. . . . . . . . . . . . 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 1118	. . . . . . . . . . . 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 458	. . . . . . . . . . 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 715	. . . . . . . . . 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 11114	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
79		recn 11114	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
80	58	recnd 11158	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ)
81		adddir 11121	. . . . . . . . . . . . . 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 1160	. . . . . . . . . . . . 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 1118	. . . . . . . . . . . 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 5108	. . . . . . . . . . 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 714	. . . . . . . . . 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 294	. . . . . . . . 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 47	. . . . . . . 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 3189	. . . . . . 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 11107	. . . . . . . . 9 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
90		breq2 5100	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
91		fveq2 6832	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
92	91	oveq1d 7371	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
93		fvoveq1 7379	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
94	93	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
95	92, 94	breq12d 5109	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
96		fveq2 6832	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
97	96	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
98		oveq2 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
99	98	fveq2d 6836	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
100	99	oveq2d 7372	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
101	97, 100	breq12d 5109	. . . . . . . . . . . . . 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 3216	. . . . . . . . . . . . 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 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑓 + 𝑔) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))))
104	103	breq2d 5108	. . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . 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 283	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 3574	. . . . . . . . . 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 412	. . . . . . . . 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 17	. . . . . . . 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 481	. . . . . . 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 608	. . . . . 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 242	. . . . 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 3191	. . . 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 242	. . 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 1116	. 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 209	1 ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058   ∩ cin 3898   class class class wbr 5096   ↦ cmpt 5177  ‘cfv 6490  ℩crio 7312  (class class class)co 7356  ℂcc 11022  ℝcr 11023  0cc0 11024   + caddc 11027   · cmul 11029   < clt 11164   ≤ cle 11165   ℋchba 30943   +_ℎ cva 30944  norm_ℎcno 30947   S_ℋ csh 30952   +_ℋ cph 30955  0_ℋc0h 30959

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102  ax-hilex 31023  ax-hfvadd 31024  ax-hvcom 31025  ax-hvass 31026  ax-hv0cl 31027  ax-hvaddid 31028  ax-hfvmul 31029  ax-hvmulid 31030  ax-hvmulass 31031  ax-hvdistr1 31032  ax-hvdistr2 31033  ax-hvmul0 31034  ax-hfi 31103  ax-his1 31106  ax-his3 31108  ax-his4 31109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-sup 9343  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-rp 12904  df-seq 13923  df-exp 13983  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-grpo 30517  df-ablo 30569  df-hnorm 30992  df-hvsub 30995  df-sh 31231  df-ch0 31277  df-shs 31332

This theorem is referenced by: (None)