Theorem xmspropd 14713

Description: Property deduction for an extended 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
xmspropd	⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))

Proof of Theorem xmspropd

Step	Hyp	Ref	Expression
1		xmspropd.1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		xmspropd.2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3	1, 2	eqtr3d 2231	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		xmspropd.4	. . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
5	3, 4	tpspropd 14272	. . 3 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
6		xmspropd.3	. . . . . . 7 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
7	1	sqxpeqd 4689	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
8	7	reseq2d 4946	. . . . . . 7 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	6, 8	eqtr3d 2231	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
10	2	sqxpeqd 4689	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
11	10	reseq2d 4946	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	9, 11	eqtr3d 2231	. . . . 5 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
13	12	fveq2d 5562	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
14	4, 13	eqeq12d 2211	. . 3 ⊢ (𝜑 → ((TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
15	5, 14	anbi12d 473	. 2 ⊢ (𝜑 → ((𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))))
16		eqid 2196	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
17		eqid 2196	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2196	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
19	16, 17, 18	isxms 14687	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))))
20		eqid 2196	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
21		eqid 2196	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2196	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
23	20, 21, 22	isxms 14687	. 2 ⊢ (𝐿 ∈ ∞MetSp ↔ (𝐿 ∈ TopSp ∧ (TopOpen‘𝐿) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
24	15, 19, 23	3bitr4g 223	1 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167   × cxp 4661   ↾ cres 4665  ‘cfv 5258  Basecbs 12678  distcds 12764  TopOpenctopn 12911  MetOpencmopn 14097  TopSpctps 14266  ∞MetSpcxms 14572

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-ndx 12681  df-slot 12682  df-base 12684  df-tset 12774  df-rest 12912  df-topn 12913  df-top 14234  df-topon 14247  df-topsp 14267  df-xms 14575

This theorem is referenced by:  mspropd  14714