Theorem metf1o 33551

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 6137	. . . . . . 7 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → 𝐹:𝑌⟶𝑋)
2		ffvelrn 6357	. . . . . . . . 9 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (𝐹‘𝑥) ∈ 𝑋)
3	2	ex 450	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → (𝑥 ∈ 𝑌 → (𝐹‘𝑥) ∈ 𝑋))
4		ffvelrn 6357	. . . . . . . . 9 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) ∈ 𝑋)
5	4	ex 450	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → (𝑦 ∈ 𝑌 → (𝐹‘𝑦) ∈ 𝑋))
6	3, 5	anim12d 586	. . . . . . 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 22137	. . . . . . 7 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹‘𝑥) ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ 𝑋) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ)
9	8	3expib 1268	. . . . . 6 ⊢ (𝑀 ∈ (Met‘𝑋) → (((𝐹‘𝑥) ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ 𝑋) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ))
10	7, 9	sylan9r 690	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ))
11	10	3adant1 1079	. . . 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 2970	. . 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	fmpt2 7237	. . 3 ⊢ (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) ∈ ℝ ↔ 𝑁:(𝑌 × 𝑌)⟶ℝ)
15	12, 14	sylib 208	. 2 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁:(𝑌 × 𝑌)⟶ℝ)
16		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑢 → (𝐹‘𝑥) = (𝐹‘𝑢))
17	16	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑀(𝐹‘𝑦)))
18		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑣 → (𝐹‘𝑦) = (𝐹‘𝑣))
19	18	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑢)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
20		ovex 6678	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢)𝑀(𝐹‘𝑣)) ∈ V
21	17, 19, 13, 20	ovmpt2 6796	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝑢𝑁𝑣) = ((𝐹‘𝑢)𝑀(𝐹‘𝑣)))
22	21	eqeq1d 2624	. . . . . . . 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		ffvelrn 6357	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑢 ∈ 𝑌) → (𝐹‘𝑢) ∈ 𝑋)
25	24	ex 450	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → (𝑢 ∈ 𝑌 → (𝐹‘𝑢) ∈ 𝑋))
26		ffvelrn 6357	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑣 ∈ 𝑌) → (𝐹‘𝑣) ∈ 𝑋)
27	26	ex 450	. . . . . . . . . . . 12 ⊢ (𝐹:𝑌⟶𝑋 → (𝑣 ∈ 𝑌 → (𝐹‘𝑣) ∈ 𝑋))
28	25, 27	anim12d 586	. . . . . . . . . . 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 445	. . . . . . . . 9 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋))
31	30	adantll 750	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋))
32		meteq0 22144	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋) → (((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
33	32	3expb 1266	. . . . . . . . 9 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹‘𝑢) ∈ 𝑋 ∧ (𝐹‘𝑣) ∈ 𝑋)) → (((𝐹‘𝑢)𝑀(𝐹‘𝑣)) = 0 ↔ (𝐹‘𝑢) = (𝐹‘𝑣)))
34	33	adantlr 751	. . . . . . . 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 487	. . . . . . 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 6136	. . . . . . . . 9 ⊢ (𝐹:𝑌–1-1-onto→𝑋 → 𝐹:𝑌–1-1→𝑋)
37		f1fveq 6519	. . . . . . . . 9 ⊢ ((𝐹:𝑌–1-1→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
38	36, 37	sylan 488	. . . . . . . 8 ⊢ ((𝐹:𝑌–1-1-onto→𝑋 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ 𝑢 = 𝑣))
39	38	adantll 750	. . . . . . 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 294	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣))
41		ffvelrn 6357	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑌⟶𝑋 ∧ 𝑤 ∈ 𝑌) → (𝐹‘𝑤) ∈ 𝑋)
42	41	ex 450	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑌⟶𝑋 → (𝑤 ∈ 𝑌 → (𝐹‘𝑤) ∈ 𝑋))
43	28, 42	anim12d 586	. . . . . . . . . . . . 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 445	. . . . . . . . . . 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 750	. . . . . . . . . 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 22146	. . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . 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 1266	. . . . . . . . . . . . 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 469	. . . . . . . . . . . 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 446	. . . . . . . . . . 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 751	. . . . . . . . . 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 487	. . . . . . . . 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 680	. . . . . . . 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 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
57	56	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑦)))
58		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (𝐹‘𝑦) = (𝐹‘𝑢))
59	58	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑢 → ((𝐹‘𝑤)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
60		ovex 6678	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤)𝑀(𝐹‘𝑢)) ∈ V
61	57, 59, 13, 60	ovmpt2 6796	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑢 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
62	61	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
63	62	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑢) = ((𝐹‘𝑤)𝑀(𝐹‘𝑢)))
64	18	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑤)𝑀(𝐹‘𝑦)) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
65		ovex 6678	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑤)𝑀(𝐹‘𝑣)) ∈ V
66	57, 64, 13, 65	ovmpt2 6796	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
67	66	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
68	67	adantll 750	. . . . . . . . . . 11 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → (𝑤𝑁𝑣) = ((𝐹‘𝑤)𝑀(𝐹‘𝑣)))
69	63, 68	oveq12d 6668	. . . . . . . . . 10 ⊢ (((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) ∧ 𝑤 ∈ 𝑌) → ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) = (((𝐹‘𝑤)𝑀(𝐹‘𝑢)) + ((𝐹‘𝑤)𝑀(𝐹‘𝑣))))
70	55, 69	breq12d 4666	. . . . . . . . 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 750	. . . . . . . 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 247	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
73	72	ralrimiva 2966	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ∀𝑤 ∈ 𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
74	40, 73	jca 554	. . . . 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 1217	. . . 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 450	. . 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 2970	. 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 22128	. . 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 1082	. 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 957	1 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653   × cxp 5112  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ℝcr 9935  0cc0 9936   + caddc 9939   ≤ cle 10075  Metcme 19732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-xadd 11947  df-xmet 19739  df-met 19740

This theorem is referenced by: (None)