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Mirrors  >  Home  >  ILE Home  >  Th. List  >  blssexps GIF version
Theorem blssexps 14014
Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
Assertion
Ref Expression
blssexps ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
Distinct variable groups:   π‘₯,π‘Ÿ,𝐴   𝐷,π‘Ÿ,π‘₯   𝑃,π‘Ÿ,π‘₯   𝑋,π‘Ÿ,π‘₯
Proof of Theorem blssexps
StepHypRef Expression
1 blssps 14012 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)
2 sstr 3165 . . . . . . . . 9 (((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)
32expcom 116 . . . . . . . 8 (π‘₯ βŠ† 𝐴 β†’ ((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
43reximdv 2578 . . . . . . 7 (π‘₯ βŠ† 𝐴 β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
51, 4syl5com 29 . . . . . 6 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ (π‘₯ βŠ† 𝐴 β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
653expa 1203 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·)) ∧ 𝑃 ∈ π‘₯) β†’ (π‘₯ βŠ† 𝐴 β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
76expimpd 363 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·)) β†’ ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
87adantlr 477 . . 3 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘₯ ∈ ran (ballβ€˜π·)) β†’ ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
98rexlimdva 2594 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
10 simpll 527 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
11 simplr 528 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝑃 ∈ 𝑋)
12 rpxr 9663 . . . . . 6 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
1312ad2antrl 490 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ*)
14 blelrnps 14004 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
1510, 11, 13, 14syl3anc 1238 . . . 4 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
16 simprl 529 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ+)
17 blcntrps 14000 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
1810, 11, 16, 17syl3anc 1238 . . . 4 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
19 simprr 531 . . . 4 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)
20 eleq2 2241 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ)))
21 sseq1 3180 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (π‘₯ βŠ† 𝐴 ↔ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
2220, 21anbi12d 473 . . . . 5 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ ((𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ (𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ) ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)))
2322rspcev 2843 . . . 4 (((𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·) ∧ (𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ) ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))
2415, 18, 19, 23syl12anc 1236 . . 3 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴))
2524rexlimdvaa 2595 . 2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴 β†’ βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴)))
269, 25impbid 129 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) β†’ (βˆƒπ‘₯ ∈ ran (ballβ€˜π·)(𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝐴) ↔ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   βŠ† wss 3131  ran crn 4629  β€˜cfv 5218  (class class class)co 5877  β„*cxr 7993  β„+crp 9655  PsMetcpsmet 13524  ballcbl 13527
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-psmet 13532  df-bl 13535
This theorem is referenced by: (None)
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