Theorem blin 21448

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
blin	|- ( ( ( 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 ) if ( R <_ S , R , S ) ) )

Proof of Theorem blin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xmetcl 21358	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ x e. X ) -> ( P D x ) e. RR* )
2	1	3expa 1209	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. X ) -> ( P D x ) e. RR* )
3	2	adantlr 722	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> ( P D x ) e. RR* )
4		simplrl 771	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> R e. RR* )
5		simplrr 772	. . . . 5 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> S e. RR* )
6		xrltmin 11484	. . . . 5 |- ( ( ( P D x ) e. RR* /\ R e. RR* /\ S e. RR* ) -> ( ( P D x ) < if ( R <_ S , R , S ) <-> ( ( P D x ) < R /\ ( P D x ) < S ) ) )
7	3, 4, 5, 6	syl3anc 1269	. . . 4 |- ( ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) /\ x e. X ) -> ( ( P D x ) < if ( R <_ S , R , S ) <-> ( ( P D x ) < R /\ ( P D x ) < S ) ) )
8	7	pm5.32da 647	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( ( x e. X /\ ( P D x ) < if ( R <_ S , R , S ) ) <-> ( x e. X /\ ( ( P D x ) < R /\ ( P D x ) < S ) ) ) )
9		ifcl 3925	. . . 4 |- ( ( R e. RR* /\ S e. RR* ) -> if ( R <_ S , R , S ) e. RR* )
10		elbl 21415	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ if ( R <_ S , R , S ) e. RR* ) -> ( x e. ( P ( ball ` D ) if ( R <_ S , R , S ) ) <-> ( x e. X /\ ( P D x ) < if ( R <_ S , R , S ) ) ) )
11	10	3expa 1209	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ if ( R <_ S , R , S ) e. RR* ) -> ( x e. ( P ( ball ` D ) if ( R <_ S , R , S ) ) <-> ( x e. X /\ ( P D x ) < if ( R <_ S , R , S ) ) ) )
12	9, 11	sylan2 477	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( R e. RR* /\ S e. RR* ) ) -> ( x e. ( P ( ball ` D ) if ( R <_ S , R , S ) ) <-> ( x e. X /\ ( P D x ) < if ( R <_ S , R , S ) ) ) )
13		elbl 21415	. . . . . . 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 1209	. . . . . 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 724	. . . . 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 21415	. . . . . . 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 1209	. . . . . 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 723	. . . . 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 718	. . . 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 3619	. . . 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 838	. . . 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 292	. . 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 290	. 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 ) if ( R <_ S , R , S ) ) ) )
24	23	eqrdv 2451	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 ) if ( R <_ S , R , S ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   i^i cin 3405   ifcif 3883   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   RR*cxr 9679   < clt 9680   <_ cle 9681   *Metcxmt 18967   ballcbl 18969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-pre-lttri 9618  ax-pre-lttrn 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-psmet 18974  df-xmet 18975  df-bl 18977

This theorem is referenced by:  blin2  21456