Theorem cdj3i 32518

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 32511	. . 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 32514	. . . 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 32517	. . . 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 5102	. . . . . . . . 9 ⊢ (𝑣 = 𝑓 → (0 < 𝑣 ↔ 0 < 𝑓))
14		oveq1 7365	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑓 → (𝑣 · (normℎ‘𝑢)) = (𝑓 · (normℎ‘𝑢)))
15	14	breq2d 5110	. . . . . . . . . 10 ⊢ (𝑣 = 𝑓 → ((normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑆‘𝑢)) ≤ (𝑓 · (normℎ‘𝑢))))
16	15	ralbidv 3159	. . . . . . . . 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 3215	. . . . . . 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 5102	. . . . . . . . 9 ⊢ (𝑣 = 𝑔 → (0 < 𝑣 ↔ 0 < 𝑔))
21		oveq1 7365	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑔 → (𝑣 · (normℎ‘𝑢)) = (𝑔 · (normℎ‘𝑢)))
22	21	breq2d 5110	. . . . . . . . . 10 ⊢ (𝑣 = 𝑔 → ((normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢)) ↔ (normℎ‘(𝑇‘𝑢)) ≤ (𝑔 · (normℎ‘𝑢))))
23	22	ralbidv 3159	. . . . . . . . 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 3215	. . . . . . 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 3208	. . . . 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 11625	. . . . . . . . 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 31441	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑡 +ℎ ℎ) ∈ (𝐴 +ℋ 𝐵))
35		2fveq3 6839	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑆‘𝑢)) = (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))))
36		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘𝑢) = (normℎ‘(𝑡 +ℎ ℎ)))
37	36	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑓 · (normℎ‘𝑢)) = (𝑓 · (normℎ‘(𝑡 +ℎ ℎ))))
38	35, 37	breq12d 5111	. . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (normℎ‘(𝑇‘𝑢)) = (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))))
41	36	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑢 = (𝑡 +ℎ ℎ) → (𝑔 · (normℎ‘𝑢)) = (𝑔 · (normℎ‘(𝑡 +ℎ ℎ))))
42	40, 41	breq12d 5111	. . . . . . . . . . . . 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 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 31291	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝐴 → 𝑡 ∈ ℋ)
48		normcl 31202	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ℋ → (normℎ‘𝑡) ∈ ℝ)
49	47, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝐴 → (normℎ‘𝑡) ∈ ℝ)
50	2	sheli 31291	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ 𝐵 → ℎ ∈ ℋ)
51		normcl 31202	. . . . . . . . . . . . . . 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 31089	. . . . . . . . . . . . . . . 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 31202	. . . . . . . . . . . . . . 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 11113	. . . . . . . . . . . . . 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 11113	. . . . . . . . . . . . . 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 11621	. . . . . . . . . . . 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 32512	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝑡 +ℎ ℎ)) = 𝑡)
69	68	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑆‘(𝑡 +ℎ ℎ))) = (normℎ‘𝑡))
70	69	breq1d 5108	. . . . . . . . . . . . . 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 32515	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝑡 +ℎ ℎ)) = ℎ)
72	71	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (normℎ‘(𝑇‘(𝑡 +ℎ ℎ))) = (normℎ‘ℎ))
73	72	breq1d 5108	. . . . . . . . . . . . . 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 11118	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ ℝ → 𝑓 ∈ ℂ)
79		recn 11118	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ ℝ → 𝑔 ∈ ℂ)
80	58	recnd 11162	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (normℎ‘(𝑡 +ℎ ℎ)) ∈ ℂ)
81		adddir 11125	. . . . . . . . . . . . . 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 5110	. . . . . . . . . . 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 3191	. . . . . . 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 11111	. . . . . . . . 9 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 + 𝑔) ∈ ℝ)
90		breq2 5102	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑓 + 𝑔) → (0 < 𝑣 ↔ 0 < (𝑓 + 𝑔)))
91		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (normℎ‘𝑥) = (normℎ‘𝑡))
92	91	oveq1d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → ((normℎ‘𝑥) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘𝑦)))
93		fvoveq1 7381	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑡 → (normℎ‘(𝑥 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ 𝑦)))
94	93	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑡 → (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))))
95	92, 94	breq12d 5111	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦))) ↔ ((normℎ‘𝑡) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦)))))
96		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ℎ → (normℎ‘𝑦) = (normℎ‘ℎ))
97	96	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → ((normℎ‘𝑡) + (normℎ‘𝑦)) = ((normℎ‘𝑡) + (normℎ‘ℎ)))
98		oveq2 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ℎ → (𝑡 +ℎ 𝑦) = (𝑡 +ℎ ℎ))
99	98	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ℎ → (normℎ‘(𝑡 +ℎ 𝑦)) = (normℎ‘(𝑡 +ℎ ℎ)))
100	99	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ℎ → (𝑣 · (normℎ‘(𝑡 +ℎ 𝑦))) = (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))))
101	97, 100	breq12d 5111	. . . . . . . . . . . . . 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 3218	. . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑓 + 𝑔) → (𝑣 · (normℎ‘(𝑡 +ℎ ℎ))) = ((𝑓 + 𝑔) · (normℎ‘(𝑡 +ℎ ℎ))))
104	103	breq2d 5110	. . . . . . . . . . . . . 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 3200	. . . . . . . . . . . . 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 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 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 3193	. . . 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 3051  ∃wrex 3060   ∩ cin 3900   class class class wbr 5098   ↦ cmpt 5179  ‘cfv 6492  ℩crio 7314  (class class class)co 7358  ℂcc 11026  ℝcr 11027  0cc0 11028   + caddc 11031   · cmul 11033   < clt 11168   ≤ cle 11169   ℋchba 30996   +_ℎ cva 30997  norm_ℎcno 31000   S_ℋ csh 31005   +_ℋ cph 31008  0_ℋc0h 31012

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106  ax-hilex 31076  ax-hfvadd 31077  ax-hvcom 31078  ax-hvass 31079  ax-hv0cl 31080  ax-hvaddid 31081  ax-hfvmul 31082  ax-hvmulid 31083  ax-hvmulass 31084  ax-hvdistr1 31085  ax-hvdistr2 31086  ax-hvmul0 31087  ax-hfi 31156  ax-his1 31159  ax-his3 31161  ax-his4 31162

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-sup 9347  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-z 12491  df-uz 12754  df-rp 12908  df-seq 13927  df-exp 13987  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-grpo 30570  df-ablo 30622  df-hnorm 31045  df-hvsub 31048  df-sh 31284  df-ch0 31330  df-shs 31385

This theorem is referenced by: (None)