Theorem blininf 13218

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 13146	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
2	1	3expa 1198	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
3	2	adantlr 474	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> ( P D x ) e. RR* )
4		simplrl 530	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> R e. RR* )
5		simplrr 531	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> S e. RR* )
6		xrltmininf 11233	. . . . 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 1233	. . . 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 449	. . 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 11229	. . . 4 |- ( ( R e. RR* /\ S e. RR* ) -> inf ( { R , S } , RR* , < ) e. RR* )
10		elbl 13185	. . . . 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 1198	. . . 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 13185	. . . . . . 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 1198	. . . . . 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 476	. . . . 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 13185	. . . . . . 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 1198	. . . . . 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 475	. . . . 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 470	. . . 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 3310	. . . 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 585	. . . 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 2168	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 1348   e. wcel 2141   i^i cin 3120   {cpr 3584   class class class wbr 3989   ` cfv 5198  (class class class)co 5853  infcinf 6960   RR*cxr 7953   < clt 7954   *Metcxmet 12774   ballcbl 12776

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-xneg 9729  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-psmet 12781  df-xmet 12782  df-bl 12784

This theorem is referenced by:  blin2  13226