Theorem metf1o 36428

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 6820	. . . . . . 7 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → 𝐹:𝑌⟶𝑋)
2		ffvelcdm 7068	. . . . . . . . 9 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (𝐹‘𝑥) ∈ 𝑋)
3	2	ex 413	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → (𝑥 ∈ 𝑌 → (𝐹‘𝑥) ∈ 𝑋))
4		ffvelcdm 7068	. . . . . . . . 9 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) ∈ 𝑋)
5	4	ex 413	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → (𝑦 ∈ 𝑌 → (𝐹‘𝑦) ∈ 𝑋))
6	3, 5	anim12d 609	. . . . . . 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 23767	. . . . . . 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 509	. . . . 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 3197	. . 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 8036	. . 3 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ ↔ 𝑁:(𝑌 × 𝑌)⟶ℝ)
15	12, 14	sylib 217	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁:(𝑌 × 𝑌)⟶ℝ)
16		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
17	16	oveq1d 7408	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑀(𝐹‘𝑦)))
18		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
19	18	oveq2d 7409	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑢)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
20		ovex 7426	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ∈ V
21	17, 19, 13, 20	ovmpo 7551	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝑢𝑁𝑣) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
22	21	eqeq1d 2733	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0))
23	22	adantl 482	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0))
24		ffvelcdm 7068	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑢 ∈ 𝑌) → (𝐹‘𝑢) ∈ 𝑋)
25	24	ex 413	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → (𝑢 ∈ 𝑌 → (𝐹‘𝑢) ∈ 𝑋))
26		ffvelcdm 7068	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑣 ∈ 𝑌) → (𝐹‘𝑣) ∈ 𝑋)
27	26	ex 413	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → (𝑣 ∈ 𝑌 → (𝐹‘𝑣) ∈ 𝑋))
28	25, 27	anim12d 609	. . . . . . . . . . 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 407	. . . . . . . . 9 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋))
31	30	adantll 712	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋))
32		meteq0 23774	. . . . . . . . . 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 713	. . . . . . . 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 591	. . . . . . 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 6819	. . . . . . . . 9 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → 𝐹:𝑌–1-1→𝑋)
37		f1fveq 7245	. . . . . . . . 9 ⊢ ((𝐹:𝑌–1-1→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
38	36, 37	sylan 580	. . . . . . . 8 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
39	38	adantll 712	. . . . . . 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 304	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣))
41		ffvelcdm 7068	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑤 ∈ 𝑌) → (𝐹‘𝑤) ∈ 𝑋)
42	41	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → (𝑤 ∈ 𝑌 → (𝐹‘𝑤) ∈ 𝑋))
43	28, 42	anim12d 609	. . . . . . . . . . . . 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 407	. . . . . . . . . . 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 712	. . . . . . . . . 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 23776	. . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . 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 459	. . . . . . . . . . . 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 408	. . . . . . . . . . 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 713	. . . . . . . . . 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 591	. . . . . . . . 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 468	. . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑢𝑁𝑣) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
56		fveq2 6878	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
57	56	oveq1d 7408	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑦)))
58		fveq2 6878	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
59	58	oveq2d 7409	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑢 → ((𝐹‘𝑤)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
60		ovex 7426	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤)𝑀(𝐹‘𝑢)) ∈ V
61	57, 59, 13, 60	ovmpo 7551	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
62	61	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
63	62	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
64	18	oveq2d 7409	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑤)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
65		ovex 7426	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤)𝑀(𝐹‘𝑣)) ∈ V
66	57, 64, 13, 65	ovmpo 7551	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
67	66	ancoms 459	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
68	67	adantll 712	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
69	63, 68	oveq12d 7411	. . . . . . . . . 10 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) = (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
70	55, 69	breq12d 5154	. . . . . . . . 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 712	. . . . . . . 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 256	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
73	72	ralrimiva 3145	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
74	40, 73	jca 512	. . . . 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 413	. . 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 3197	. 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 23758	. . 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 711	1 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060   class class class wbr 5141   × cxp 5667  ⟶wf 6528  –1-1→wf1 6529  –1-1-onto→wf1o 6531  ‘cfv 6532  (class class class)co 7393   ∈ cmpo 7395  ℝcr 11091  0cc0 11092   + caddc 11095   ≤ cle 11231  Metcmet 20864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-mulcl 11154  ax-i2m1 11160

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-xadd 13075  df-xmet 20871  df-met 20872

This theorem is referenced by: (None)