Theorem mspropd 24500

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 24499	. . 3 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
6	1	sqxpeqd 5721	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
7	6	reseq2d 6000	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	3, 7	eqtr3d 2777	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	2	sqxpeqd 5721	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
10	9	reseq2d 6000	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11	8, 10	eqtr3d 2777	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	1, 2	eqtr3d 2777	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
13	12	fveq2d 6911	. . . 4 ⊢ (𝜑 → (Met‘(Base‘𝐾)) = (Met‘(Base‘𝐿)))
14	11, 13	eleq12d 2833	. . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
15	5, 14	anbi12d 632	. 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 2735	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
17		eqid 2735	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2735	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
19	16, 17, 18	isms 24475	. 2 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))))
20		eqid 2735	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
21		eqid 2735	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2735	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
23	20, 21, 22	isms 24475	. 2 ⊢ (𝐿 ∈ MetSp ↔ (𝐿 ∈ ∞MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (Met‘(Base‘𝐿))))
24	15, 19, 23	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   × cxp 5687   ↾ cres 5691  ‘cfv 6563  Basecbs 17245  distcds 17307  TopOpenctopn 17468  Metcmet 21368  ∞MetSpcxms 24343  MetSpcms 24344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-topon 22933  df-topsp 22955  df-xms 24346  df-ms 24347

This theorem is referenced by:  ngppropd  24666  cmspropd  25397  zhmnrg  33928