Theorem ioo2bl 15342

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 8201	. . . . 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 9434	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) / 2 ) e. RR )
4		resubcl 8486	. . . . 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 9434	. . 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 15341	. . 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 8208	. . . . 5 |- ( B e. RR -> B e. CC )
11		recn 8208	. . . . 5 |- ( A e. RR -> A e. CC )
12		addcom 8359	. . . . 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 6043	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) / 2 ) = ( ( A + B ) / 2 ) )
15	14	oveq1d 6043	. 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 9421	. . . . 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 6046	. 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 2273	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 2202   X. cxp 4729   |` cres 4733   o. ccom 4735   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   + caddc 8078   - cmin 8393   / cdiv 8895   2c2 9237   (,)cioo 10166   abscabs 11618   ballcbl 14614

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-n0 9446  df-z 9523  df-uz 9799  df-rp 9932  df-xadd 10051  df-ioo 10170  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-psmet 14619  df-xmet 14620  df-met 14621  df-bl 14622

This theorem is referenced by:  ioo2blex  15343