Theorem blininf 15083

Description: The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)

Assertion

Ref	Expression
blininf	|- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( ( P ( ball ` D ) R ) i^i ( P ( ball ` D ) S ) ) = ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) )

Proof of Theorem blininf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetcl 15011	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
2	1	3expa 1227	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
3	2	adantlr 477	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> ( P D x ) e. RR* )
4		simplrl 535	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> R e. RR* )
5		simplrr 536	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> S e. RR* )
6		xrltmininf 11767	. . . . 5 |- ( ( ( P D x ) e. RR* /\ R e. RR* /\ S e. RR* ) -> ( ( P D x ) < inf ( { R , S } , RR* , < ) <-> ( ( P D x ) < R /\ ( P D x ) < S ) ) )
7	3, 4, 5, 6	syl3anc 1271	. . . 4 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> ( ( P D x ) < inf ( { R , S } , RR* , < ) <-> ( ( P D x ) < R /\ ( P D x ) < S ) ) )
8	7	pm5.32da 452	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( ( x e. X /\ ( P D x ) < inf ( { R , S } , RR* , < ) ) <-> ( x e. X /\ ( ( P D x ) < R /\ ( P D x ) < S ) ) ) )
9		xrmincl 11763	. . . 4 |- ( ( R e. RR* /\ S e. RR* ) -> inf ( { R , S } , RR* , < ) e. RR* )
10		elbl 15050	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ inf ( { R , S } , RR* , < ) e. RR* ) -> ( x e. ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) <-> ( x e. X /\ ( P D x ) < inf ( { R , S } , RR* , < ) ) ) )
11	10	3expa 1227	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ inf ( { R , S } , RR* , < ) e. RR* ) -> ( x e. ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) <-> ( x e. X /\ ( P D x ) < inf ( { R , S } , RR* , < ) ) ) )
12	9, 11	sylan2 286	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( x e. ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) <-> ( x e. X /\ ( P D x ) < inf ( { R , S } , RR* , < ) ) ) )
13		elbl 15050	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
14	13	3expa 1227	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ R e. RR* ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
15	14	adantrr 479	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( x e. ( P ( ball ` D ) R ) <-> ( x e. X /\ ( P D x ) < R ) ) )
16		elbl 15050	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ S e. RR* ) -> ( x e. ( P ( ball ` D ) S ) <-> ( x e. X /\ ( P D x ) < S ) ) )
17	16	3expa 1227	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ S e. RR* ) -> ( x e. ( P ( ball ` D ) S ) <-> ( x e. X /\ ( P D x ) < S ) ) )
18	17	adantrl 478	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( x e. ( P ( ball ` D ) S ) <-> ( x e. X /\ ( P D x ) < S ) ) )
19	15, 18	anbi12d 473	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( ( x e. ( P ( ball ` D ) R ) /\ x e. ( P ( ball ` D ) S ) ) <-> ( ( x e. X /\ ( P D x ) < R ) /\ ( x e. X /\ ( P D x ) < S ) ) ) )
20		elin 3387	. . . 4 |- ( x e. ( ( P ( ball ` D ) R ) i^i ( P ( ball ` D ) S ) ) <-> ( x e. ( P ( ball ` D ) R ) /\ x e. ( P ( ball ` D ) S ) ) )
21		anandi 592	. . . 4 |- ( ( x e. X /\ ( ( P D x ) < R /\ ( P D x ) < S ) ) <-> ( ( x e. X /\ ( P D x ) < R ) /\ ( x e. X /\ ( P D x ) < S ) ) )
22	19, 20, 21	3bitr4g 223	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( x e. ( ( P ( ball ` D ) R ) i^i ( P ( ball ` D ) S ) ) <-> ( x e. X /\ ( ( P D x ) < R /\ ( P D x ) < S ) ) ) )
23	8, 12, 22	3bitr4rd 221	. 2 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( x e. ( ( P ( ball ` D ) R ) i^i ( P ( ball ` D ) S ) ) <-> x e. ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) ) )
24	23	eqrdv 2227	1 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( ( P ( ball ` D ) R ) i^i ( P ( ball ` D ) S ) ) = ( P ( ball ` D )inf ( { R , S } , RR* , < ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   i^i cin 3196   {cpr 3667   class class class wbr 4082   ` cfv 5314  (class class class)co 5994  infcinf 7138   RR*cxr 8168   < clt 8169   *Metcxmet 14485   ballcbl 14487

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-isom 5323  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-map 6787  df-sup 7139  df-inf 7140  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-rp 9838  df-xneg 9956  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-psmet 14492  df-xmet 14493  df-bl 14495

This theorem is referenced by:  blin2  15091