Theorem ioo2bl 15635

Description: An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Hypothesis

Ref	Expression
remet.1	|- D = ( ( abs o. - ) |` ( RR X. RR ) )

Assertion

Ref	Expression
ioo2bl	|- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) )

Proof of Theorem ioo2bl

Step	Hyp	Ref	Expression
1		readdcl 8299	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( B + A ) e. RR )
2	1	ancoms 268	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B + A ) e. RR )
3	2	rehalfcld 9535	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) / 2 ) e. RR )
4		resubcl 8584	. . . . 5 |- ( ( B e. RR /\ A e. RR ) -> ( B - A ) e. RR )
5	4	ancoms 268	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B - A ) e. RR )
6	5	rehalfcld 9535	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B - A ) / 2 ) e. RR )
7		remet.1	. . . 4 |- D = ( ( abs o. - ) |` ( RR X. RR ) )
8	7	bl2ioo 15634	. . 3 |- ( ( ( ( B + A ) / 2 ) e. RR /\ ( ( B - A ) / 2 ) e. RR ) -> ( ( ( B + A ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) = ( ( ( ( B + A ) / 2 ) - ( ( B - A ) / 2 ) ) (,) ( ( ( B + A ) / 2 ) + ( ( B - A ) / 2 ) ) ) )
9	3, 6, 8	syl2anc 415	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) = ( ( ( ( B + A ) / 2 ) - ( ( B - A ) / 2 ) ) (,) ( ( ( B + A ) / 2 ) + ( ( B - A ) / 2 ) ) ) )
10		recn 8306	. . . . 5 |- ( B e. RR -> B e. CC )
11		recn 8306	. . . . 5 |- ( A e. RR -> A e. CC )
12		addcom 8457	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> ( B + A ) = ( A + B ) )
13	10, 11, 12	syl2anr 290	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( B + A ) = ( A + B ) )
14	13	oveq1d 6094	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) / 2 ) = ( ( A + B ) / 2 ) )
15	14	oveq1d 6094	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) )
16		halfaddsub 9522	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> ( ( ( ( B + A ) / 2 ) + ( ( B - A ) / 2 ) ) = B /\ ( ( ( B + A ) / 2 ) - ( ( B - A ) / 2 ) ) = A ) )
17	10, 11, 16	syl2anr 290	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( B + A ) / 2 ) + ( ( B - A ) / 2 ) ) = B /\ ( ( ( B + A ) / 2 ) - ( ( B - A ) / 2 ) ) = A ) )
18	17	simprd 114	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) / 2 ) - ( ( B - A ) / 2 ) ) = A )
19	17	simpld 112	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) / 2 ) + ( ( B - A ) / 2 ) ) = B )
20	18, 19	oveq12d 6097	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( B + A ) / 2 ) - ( ( B - A ) / 2 ) ) (,) ( ( ( B + A ) / 2 ) + ( ( B - A ) / 2 ) ) ) = ( A (,) B ) )
21	9, 15, 20	3eqtr3rd 2280	1 |- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   X. cxp 4770   |` cres 4774   o. ccom 4776   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   + caddc 8176   - cmin 8491   / cdiv 8996   2c2 9338   (,)cioo 10273   abscabs 11746   ballcbl 14858

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-map 6918  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-rp 10038  df-xadd 10158  df-ioo 10277  df-seqfrec 10868  df-exp 10959  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-psmet 14863  df-xmet 14864  df-met 14865  df-bl 14866

This theorem is referenced by:  ioo2blex  15636