Theorem blininf 13064

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 12992	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
2	1	3expa 1193	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
3	2	adantlr 469	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> ( P D x ) e. RR* )
4		simplrl 525	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> R e. RR* )
5		simplrr 526	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> S e. RR* )
6		xrltmininf 11211	. . . . 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 1228	. . . 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 448	. . 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 11207	. . . 4 |- ( ( R e. RR* /\ S e. RR* ) -> inf ( { R , S } , RR* , < ) e. RR* )
10		elbl 13031	. . . . 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 1193	. . . 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 284	. . 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 13031	. . . . . . 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 1193	. . . . . 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 471	. . . . 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 13031	. . . . . . 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 1193	. . . . . 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 470	. . . . 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 465	. . . 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 3305	. . . 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 580	. . . 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 222	. . 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 220	. 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 2163	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 103   <-> wb 104   = wceq 1343   e. wcel 2136   i^i cin 3115   {cpr 3577   class class class wbr 3982   ` cfv 5188  (class class class)co 5842  infcinf 6948   RR*cxr 7932   < clt 7933   *Metcxmet 12620   ballcbl 12622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-xneg 9708  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-psmet 12627  df-xmet 12628  df-bl 12630

This theorem is referenced by:  blin2  13072