Theorem mspropd 15195

Description: Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
xmspropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
xmspropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
xmspropd.3	⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
xmspropd.4	⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Assertion

Ref	Expression
mspropd	⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))

Proof of Theorem mspropd

Step	Hyp	Ref	Expression
1		xmspropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		xmspropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		xmspropd.3	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
4		xmspropd.4	. . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
5	1, 2, 3, 4	xmspropd 15194	. . 3 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
6	1	sqxpeqd 4749	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
7	6	reseq2d 5011	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	3, 7	eqtr3d 2264	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	2	sqxpeqd 4749	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
10	9	reseq2d 5011	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11	8, 10	eqtr3d 2264	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	1, 2	eqtr3d 2264	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
13	12	fveq2d 5639	. . . 4 ⊢ (𝜑 → (Met‘(Base‘𝐾)) = (Met‘(Base‘𝐿)))
14	11, 13	eleq12d 2300	. . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
15	5, 14	anbi12d 473	. 2 ⊢ (𝜑 → ((𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿)))))
16		eqid 2229	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
17		eqid 2229	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2229	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
19	16, 17, 18	isms 15170	. 2 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))))
20		eqid 2229	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
21		eqid 2229	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2229	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
23	20, 21, 22	isms 15170	. 2 ⊢ (𝐿 ∈ MetSp ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
24	15, 19, 23	3bitr4g 223	1 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   × cxp 4721   ↾ cres 4725  ‘cfv 5324  Basecbs 13075  distcds 13162  TopOpenctopn 13316  Metcmet 14544  ∞MetSpcxms 15053  MetSpcms 15054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8116  ax-resscn 8117  ax-1re 8119  ax-addrcl 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-ndx 13078  df-slot 13079  df-base 13081  df-tset 13172  df-rest 13317  df-topn 13318  df-top 14715  df-topon 14728  df-topsp 14748  df-xms 15056  df-ms 15057

This theorem is referenced by: (None)