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Theorem maxabslemval 11172
Description: Lemma for maxabs 11173. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)
Assertion
Ref Expression
maxabslemval |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x /\ A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) ) )
Distinct variable groups:   x, A, z   x, B, z
Proof of Theorem maxabslemval
StepHypRef Expression
1 readdcl 7900 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
2 simpl 108 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
32recnd 7948 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
4 simpr 109 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
54recnd 7948 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
63, 5subcld 8230 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. CC )
76abscld 11145 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) e. RR )
81, 7readdcld 7949 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
98rehalfcld 9124 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
10 vex 2733 . . . . 5 |- x e. _V
1110elpr 3604 . . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
12 maxabsle 11168 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
132, 9, 12lensymd 8041 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A )
14 breq2 3993 . . . . . . 7 |- ( x = A -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
1514notbid 662 . . . . . 6 |- ( x = A -> ( -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
1613, 15syl5ibrcom 156 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( x = A -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
17 maxabsle 11168 . . . . . . . . 9 |- ( ( B e. RR /\ A e. RR ) -> B <_ ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) )
1817ancoms 266 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) )
195, 3addcomd 8070 . . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( B + A ) = ( A + B ) )
205, 3abssubd 11157 . . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( B - A ) ) = ( abs ` ( A - B ) ) )
2119, 20oveq12d 5871 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) + ( abs ` ( B - A ) ) ) = ( ( A + B ) + ( abs ` ( A - B ) ) ) )
2221oveq1d 5868 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
2318, 22breqtrd 4015 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
244, 9, 23lensymd 8041 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B )
25 breq2 3993 . . . . . . 7 |- ( x = B -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
2625notbid 662 . . . . . 6 |- ( x = B -> ( -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
2724, 26syl5ibrcom 156 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( x = B -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
2816, 27jaod 712 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( x = A \/ x = B ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
2911, 28syl5bi 151 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( x e. { A , B } -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
3029ralrimiv 2542 . 2 |- ( ( A e. RR /\ B e. RR ) -> A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x )
31 prid1g 3687 . . . . . . 7 |- ( A e. RR -> A e. { A , B } )
3231ad4antr 491 . . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < A ) -> A e. { A , B } )
33 breq2 3993 . . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
3433rspcev 2834 . . . . . 6 |- ( ( A e. { A , B } /\ x < A ) -> E. z e. { A , B } x < z )
3532, 34sylancom 418 . . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < A ) -> E. z e. { A , B } x < z )
36 prid2g 3688 . . . . . . 7 |- ( B e. RR -> B e. { A , B } )
3736ad4antlr 492 . . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < B ) -> B e. { A , B } )
38 breq2 3993 . . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
3938rspcev 2834 . . . . . 6 |- ( ( B e. { A , B } /\ x < B ) -> E. z e. { A , B } x < z )
4037, 39sylancom 418 . . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) /\ x < B ) -> E. z e. { A , B } x < z )
412ad2antrr 485 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A e. RR )
424ad2antrr 485 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> B e. RR )
43 simplr 525 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> x e. RR )
44 simpr 109 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
4541, 42, 43, 44maxabslemlub 11171 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( x < A \/ x < B ) )
4635, 40, 45mpjaodan 793 . . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> E. z e. { A , B } x < z )
4746ex 114 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) -> ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) )
4847ralrimiva 2543 . 2 |- ( ( A e. RR /\ B e. RR ) -> A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) )
499, 30, 483jca 1172 1 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x /\ A. x e. RR ( x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } x < z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   {cpr 3584   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RRcr 7773   + caddc 7777   < clt 7954   <_ cle 7955   - cmin 8090   / cdiv 8589   2c2 8929   abscabs 10961
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-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  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-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963
This theorem is referenced by:  maxabs  11173  maxleast  11177
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