Theorem xmspropd 14656

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 2228	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4		xmspropd.4	. . . 4 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
5	3, 4	tpspropd 14215	. . 3 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
6		xmspropd.3	. . . . . . 7 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
7	1	sqxpeqd 4686	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
8	7	reseq2d 4943	. . . . . . 7 ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
9	6, 8	eqtr3d 2228	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
10	2	sqxpeqd 4686	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
11	10	reseq2d 4943	. . . . . 6 ⊢ (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
12	9, 11	eqtr3d 2228	. . . . 5 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
13	12	fveq2d 5559	. . . 4 ⊢ (𝜑 → (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
14	4, 13	eqeq12d 2208	. . 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 2193	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
17		eqid 2193	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2193	. . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
19	16, 17, 18	isxms 14630	. 2 ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))))
20		eqid 2193	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
21		eqid 2193	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2193	. . 3 ⊢ ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
23	20, 21, 22	isxms 14630	. 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 2164   × cxp 4658   ↾ cres 4662  ‘cfv 5255  Basecbs 12621  distcds 12707  TopOpenctopn 12854  MetOpencmopn 14040  TopSpctps 14209  ∞MetSpcxms 14515

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-ndx 12624  df-slot 12625  df-base 12627  df-tset 12717  df-rest 12855  df-topn 12856  df-top 14177  df-topon 14190  df-topsp 14210  df-xms 14518

This theorem is referenced by:  mspropd  14657