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Theorem maxabslemval 11997
Description: Lemma for maxabs 11998. 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 8311 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
2 simpl 109 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A e. RR )
32recnd 8360 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A e. CC )
4 simpr 110 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B e. RR )
54recnd 8360 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> B e. CC )
63, 5subcld 8644 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. CC )
76abscld 11970 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( A - B ) ) e. RR )
81, 7readdcld 8361 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
98rehalfcld 9563 . 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR )
10 vex 2824 . . . . 5 |- x e. _V
1110elpr 3730 . . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
12 maxabsle 11993 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> A <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
132, 9, 12lensymd 8455 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A )
14 breq2 4135 . . . . . . 7 |- ( x = A -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
1514notbid 677 . . . . . 6 |- ( x = A -> ( -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < A ) )
1613, 15syl5ibrcom 157 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( x = A -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
17 maxabsle 11993 . . . . . . . . 9 |- ( ( B e. RR /\ A e. RR ) -> B <_ ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) )
1817ancoms 268 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) )
195, 3addcomd 8484 . . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( B + A ) = ( A + B ) )
205, 3abssubd 11982 . . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( abs ` ( B - A ) ) = ( abs ` ( A - B ) ) )
2119, 20oveq12d 6106 . . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( B + A ) + ( abs ` ( B - A ) ) ) = ( ( A + B ) + ( abs ` ( A - B ) ) ) )
2221oveq1d 6103 . . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( B + A ) + ( abs ` ( B - A ) ) ) / 2 ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
2318, 22breqtrd 4157 . . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> B <_ ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
244, 9, 23lensymd 8455 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B )
25 breq2 4135 . . . . . . 7 |- ( x = B -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
2625notbid 677 . . . . . 6 |- ( x = B -> ( -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < B ) )
2724, 26syl5ibrcom 157 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( x = B -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
2816, 27jaod 729 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( x = A \/ x = B ) -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
2911, 28biimtrid 152 . . 3 |- ( ( A e. RR /\ B e. RR ) -> ( x e. { A , B } -> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x ) )
3029ralrimiv 2622 . 2 |- ( ( A e. RR /\ B e. RR ) -> A. x e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < x )
31 prid1g 3815 . . . . . . 7 |- ( A e. RR -> A e. { A , B } )
3231ad4antr 498 . . . . . 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 4135 . . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
3433rspcev 2929 . . . . . 6 |- ( ( A e. { A , B } /\ x < A ) -> E. z e. { A , B } x < z )
3532, 34sylancom 424 . . . . 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 3816 . . . . . . 7 |- ( B e. RR -> B e. { A , B } )
3736ad4antlr 499 . . . . . 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 4135 . . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
3938rspcev 2929 . . . . . 6 |- ( ( B e. { A , B } /\ x < B ) -> E. z e. { A , B } x < z )
4037, 39sylancom 424 . . . . 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 492 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> A e. RR )
424ad2antrr 492 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> B e. RR )
43 simplr 533 . . . . . 6 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> x e. RR )
44 simpr 110 . . . . . 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 11996 . . . . 5 |- ( ( ( ( A e. RR /\ B e. RR ) /\ x e. RR ) /\ x < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) -> ( x < A \/ x < B ) )
4635, 40, 45mpjaodan 810 . . . 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 115 . . 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 2623 . 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 1208 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
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {cpr 3710   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   + caddc 8188   < clt 8366   <_ cle 8367   - cmin 8504   / cdiv 9010   2c2 9364   abscabs 11785
This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305
This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787
This theorem is used by:  maxabs  11998  maxleast  12002
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