Theorem cmspropd 25408

Description: Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

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

Assertion

Ref	Expression
cmspropd	⊢ (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))

Proof of Theorem cmspropd

Step	Hyp	Ref	Expression
1		cmspropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		cmspropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		cmspropd.3	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
4		cmspropd.4	. . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
5	1, 2, 3, 4	mspropd 24509	. . 3 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
6	1	sqxpeqd 5725	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
7	6	reseq2d 6004	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
8	3, 7	eqtr3d 2779	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	2	sqxpeqd 5725	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
10	9	reseq2d 6004	. . . . 5 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11	8, 10	eqtr3d 2779	. . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	1, 2	eqtr3d 2779	. . . . 5 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
13	12	fveq2d 6918	. . . 4 ⊢ (𝜑 → (CMet‘(Base‘𝐾)) = (CMet‘(Base‘𝐿)))
14	11, 13	eleq12d 2835	. . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾)) ↔ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
15	5, 14	anbi12d 632	. 2 ⊢ (𝜑 → ((𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))) ↔ (𝐿 ∈ MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿)))))
16		eqid 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
17		eqid 2737	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
18	16, 17	iscms 25404	. 2 ⊢ (𝐾 ∈ CMetSp ↔ (𝐾 ∈ MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (CMet‘(Base‘𝐾))))
19		eqid 2737	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
20		eqid 2737	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
21	19, 20	iscms 25404	. 2 ⊢ (𝐿 ∈ CMetSp ↔ (𝐿 ∈ MetSp ∧ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) ∈ (CMet‘(Base‘𝐿))))
22	15, 18, 21	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ CMetSp ↔ 𝐿 ∈ CMetSp))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2108   × cxp 5691   ↾ cres 5695  ‘cfv 6569  Basecbs 17254  distcds 17316  TopOpenctopn 17477  MetSpcms 24353  CMetccmet 25313  CMetSpccms 25391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-res 5705  df-iota 6522  df-fun 6571  df-fv 6577  df-top 22925  df-topon 22942  df-topsp 22964  df-xms 24355  df-ms 24356  df-cms 25394

This theorem is referenced by:  srabn  25419