Theorem ioo2bl 15547

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 8270	. . . . 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 9506	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) / 2 ) e. RR )
4		resubcl 8555	. . . . 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 9506	. . 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 15546	. . 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 411	. 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 8277	. . . . 5 |- ( B e. RR -> B e. CC )
11		recn 8277	. . . . 5 |- ( A e. RR -> A e. CC )
12		addcom 8428	. . . . 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 6074	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) / 2 ) = ( ( A + B ) / 2 ) )
15	14	oveq1d 6074	. 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 9493	. . . . 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 6077	. 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 2276	1 |- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   X. cxp 4753   |` cres 4757   o. ccom 4759   ` cfv 5358  (class class class)co 6059   CCcc 8142   RRcr 8143   + caddc 8147   - cmin 8462   / cdiv 8967   2c2 9309   (,)cioo 10244   abscabs 11712   ballcbl 14817

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-iinf 4716  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-1re 8238  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-mulrcl 8243  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-0lt1 8250  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-precex 8254  ax-cnre 8255  ax-pre-ltirr 8256  ax-pre-ltwlin 8257  ax-pre-lttrn 8258  ax-pre-apti 8259  ax-pre-ltadd 8260  ax-pre-mulgt0 8261  ax-pre-mulext 8262  ax-arch 8263  ax-caucvg 8264

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4719  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-1st 6348  df-2nd 6349  df-recs 6550  df-frec 6636  df-map 6898  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331  df-sub 8464  df-neg 8465  df-reap 8868  df-ap 8875  df-div 8968  df-inn 9259  df-2 9317  df-3 9318  df-4 9319  df-n0 9518  df-z 9599  df-uz 9876  df-rp 10009  df-xadd 10129  df-ioo 10248  df-seqfrec 10838  df-exp 10929  df-cj 11556  df-re 11557  df-im 11558  df-rsqrt 11713  df-abs 11714  df-psmet 14822  df-xmet 14823  df-met 14824  df-bl 14825

This theorem is referenced by:  ioo2blex  15548