Theorem metf1o 37715

Description: Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
metf1o.2	⊢ 𝑁 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)𝑀(𝐹‘𝑦)))

Assertion

Ref	Expression
metf1o	⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌))

Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦

Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem metf1o

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1of 6862	. . . . . . 7 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → 𝐹:𝑌⟶𝑋)
2		ffvelcdm 7115	. . . . . . . . 9 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (𝐹‘𝑥) ∈ 𝑋)
3	2	ex 412	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → (𝑥 ∈ 𝑌 → (𝐹‘𝑥) ∈ 𝑋))
4		ffvelcdm 7115	. . . . . . . . 9 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) ∈ 𝑋)
5	4	ex 412	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → (𝑦 ∈ 𝑌 → (𝐹‘𝑦) ∈ 𝑋))
6	3, 5	anim12d 608	. . . . . . 7 ⊢ (𝐹:𝑌⟶𝑋 → ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥) ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ 𝑋)))
7	1, 6	syl 17	. . . . . 6 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥) ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ 𝑋)))
8		metcl 24363	. . . . . . 7 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹‘𝑥) ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ 𝑋) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ)
9	8	3expib 1122	. . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → (((𝐹‘𝑥) ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ 𝑋) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ))
10	7, 9	sylan9r 508	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ))
11	10	3adant1 1130	. . . 4 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ))
12	11	ralrimivv 3206	. . 3 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ)
13		metf1o.2	. . . 4 ⊢ 𝑁 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)𝑀(𝐹‘𝑦)))
14	13	fmpo 8109	. . 3 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ ↔ 𝑁:(𝑌 × 𝑌)⟶ℝ)
15	12, 14	sylib 218	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁:(𝑌 × 𝑌)⟶ℝ)
16		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
17	16	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑀(𝐹‘𝑦)))
18		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
19	18	oveq2d 7464	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑢)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
20		ovex 7481	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ∈ V
21	17, 19, 13, 20	ovmpo 7610	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝑢𝑁𝑣) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
22	21	eqeq1d 2742	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0))
23	22	adantl 481	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0))
24		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑢 ∈ 𝑌) → (𝐹‘𝑢) ∈ 𝑋)
25	24	ex 412	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → (𝑢 ∈ 𝑌 → (𝐹‘𝑢) ∈ 𝑋))
26		ffvelcdm 7115	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑣 ∈ 𝑌) → (𝐹‘𝑣) ∈ 𝑋)
27	26	ex 412	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → (𝑣 ∈ 𝑌 → (𝐹‘𝑣) ∈ 𝑋))
28	25, 27	anim12d 608	. . . . . . . . . . 11 ⊢ (𝐹:𝑌⟶𝑋 → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)))
29	1, 28	syl 17	. . . . . . . . . 10 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)))
30	29	imp 406	. . . . . . . . 9 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋))
31	30	adantll 713	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋))
32		meteq0 24370	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) → (((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
33	32	3expb 1120	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)) → (((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
34	33	adantlr 714	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)) → (((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
35	31, 34	syldan 590	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
36		f1of1 6861	. . . . . . . . 9 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → 𝐹:𝑌–1-1→𝑋)
37		f1fveq 7299	. . . . . . . . 9 ⊢ ((𝐹:𝑌–1-1→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
38	36, 37	sylan 579	. . . . . . . 8 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
39	38	adantll 713	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
40	23, 35, 39	3bitrd 305	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣))
41		ffvelcdm 7115	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑤 ∈ 𝑌) → (𝐹‘𝑤) ∈ 𝑋)
42	41	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → (𝑤 ∈ 𝑌 → (𝐹‘𝑤) ∈ 𝑋))
43	28, 42	anim12d 608	. . . . . . . . . . . . 13 ⊢ (𝐹:𝑌⟶𝑋 → (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋)))
44	1, 43	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋)))
45	44	imp 406	. . . . . . . . . . 11 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌)) → (((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋))
46	45	adantll 713	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌)) → (((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋))
47		mettri2 24372	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹‘𝑤) ∈ 𝑋 ∧ (𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
48	47	expcom 413	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑤) ∈ 𝑋 ∧ (𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) → (𝑀 ∈ (Met‘𝑋) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))))
49	48	3expb 1120	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑤) ∈ 𝑋 ∧ ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)) → (𝑀 ∈ (Met‘𝑋) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))))
50	49	ancoms 458	. . . . . . . . . . . 12 ⊢ ((((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋) → (𝑀 ∈ (Met‘𝑋) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))))
51	50	impcom 407	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋)) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
52	51	adantlr 714	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) ∧ (𝐹‘𝑤) ∈ 𝑋)) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
53	46, 52	syldan 590	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌)) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
54	53	anassrs 467	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
55	21	adantr 480	. . . . . . . . . 10 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑢𝑁𝑣) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
56		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
57	56	oveq1d 7463	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑦)))
58		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
59	58	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑢 → ((𝐹‘𝑤)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
60		ovex 7481	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤)𝑀(𝐹‘𝑢)) ∈ V
61	57, 59, 13, 60	ovmpo 7610	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
62	61	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
63	62	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
64	18	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑤)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
65		ovex 7481	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤)𝑀(𝐹‘𝑣)) ∈ V
66	57, 64, 13, 65	ovmpo 7610	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
67	66	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
68	67	adantll 713	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
69	63, 68	oveq12d 7466	. . . . . . . . . 10 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) = (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
70	55, 69	breq12d 5179	. . . . . . . . 9 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → ((𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) ↔ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))))
71	70	adantll 713	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ((𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) ↔ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ≤ (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))))
72	54, 71	mpbird 257	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
73	72	ralrimiva 3152	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
74	40, 73	jca 511	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
75	74	3adantl1 1166	. . . 4 ⊢ (((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
76	75	ex 412	. . 3 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))))
77	76	ralrimivv 3206	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
78		ismet 24354	. . 3 ⊢ (𝑌 ∈ 𝐴 → (𝑁 ∈ (Met‘𝑌) ↔ (𝑁:(𝑌 × 𝑌)⟶ℝ ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))))
79	78	3ad2ant1 1133	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → (𝑁 ∈ (Met‘𝑌) ↔ (𝑁:(𝑌 × 𝑌)⟶ℝ ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))))
80	15, 77, 79	mpbir2and 712	1 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166   × cxp 5698  ⟶wf 6569  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  ℝcr 11183  0cc0 11184   + caddc 11187   ≤ cle 11325  Metcmet 21373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-mulcl 11246  ax-i2m1 11252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-xadd 13176  df-xmet 21380  df-met 21381

This theorem is referenced by: (None)