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Theorem blssexps 23624
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 23622 . . . . . . 7 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘₯ ∈ ran (ballβ€˜π·) ∧ 𝑃 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)
2 sstr 3934 . . . . . . . . 9 (((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ∧ π‘₯ βŠ† 𝐴) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)
32expcom 415 . . . . . . . 8 (π‘₯ βŠ† 𝐴 β†’ ((𝑃(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ β†’ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
43reximdv 3164 . . . . . . 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 31 . . . . . 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 1118 . . . . 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 455 . . . 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 713 . . 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 3149 . 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 765 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝐷 ∈ (PsMetβ€˜π‘‹))
11 simplr 767 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ 𝑃 ∈ 𝑋)
12 rpxr 12785 . . . . . 6 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
1312ad2antrl 726 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ*)
14 blelrnps 23614 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘Ÿ) ∈ ran (ballβ€˜π·))
1510, 11, 13, 14syl3anc 1371 . . . 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 769 . . . . 5 (((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴)) β†’ π‘Ÿ ∈ ℝ+)
17 blcntrps 23610 . . . . 5 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ))
1810, 11, 16, 17syl3anc 1371 . . . 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 771 . . . 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 2825 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘Ÿ)))
21 sseq1 3951 . . . . . 6 (π‘₯ = (𝑃(ballβ€˜π·)π‘Ÿ) β†’ (π‘₯ βŠ† 𝐴 ↔ (𝑃(ballβ€˜π·)π‘Ÿ) βŠ† 𝐴))
2220, 21anbi12d 632 . . . . 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 3566 . . . 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 835 . . 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 3150 . 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 211 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 setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3071   βŠ† wss 3892  ran crn 5601  β€˜cfv 6458  (class class class)co 7307  β„*cxr 11054  β„+crp 12776  PsMetcpsmet 20626  ballcbl 20629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994  ax-pre-sup 10995
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-sup 9245  df-inf 9246  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-div 11679  df-nn 12020  df-2 12082  df-n0 12280  df-z 12366  df-uz 12629  df-q 12735  df-rp 12777  df-xneg 12894  df-xadd 12895  df-xmul 12896  df-psmet 20634  df-bl 20637
This theorem is referenced by:  metustbl  23767  psmetutop  23768
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