Theorem maxleast 11981

Description: The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)

Assertion

Ref	Expression
maxleast	|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ C /\ B <_ C ) ) -> sup ( { A , B } , RR , < ) <_ C )

Proof of Theorem maxleast

Dummy variables f g x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioran 764	. . . 4 |- ( -. ( C < A \/ C < B ) <-> ( -. C < A /\ -. C < B ) )
2		simp3 1030	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		lttri3 8405	. . . . . . . . 9 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
4	3	adantl 277	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
5		maxabslemval 11976	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) )
6		3anass 1013	. . . . . . . . . . 11 |- ( ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) <-> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) )
7	5, 6	sylib 122	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) )
8		breq1 4133	. . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( x < y <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
9	8	notbid 677	. . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( -. x < y <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
10	9	ralbidv 2550	. . . . . . . . . . . 12 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( A. y e. { A , B } -. x < y <-> A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
11		breq2 4134	. . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( y < x <-> y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) ) )
12	11	imbi1d 231	. . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( ( y < x -> E. z e. { A , B } y < z ) <-> ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) )
13	12	ralbidv 2550	. . . . . . . . . . . 12 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( A. y e. RR ( y < x -> E. z e. { A , B } y < z ) <-> A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) )
14	10, 13	anbi12d 477	. . . . . . . . . . 11 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) <-> ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) )
15	14	rspcev 2929	. . . . . . . . . 10 |- ( ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) e. RR /\ ( A. y e. { A , B } -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y /\ A. y e. RR ( y < ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> E. z e. { A , B } y < z ) ) ) -> E. x e. RR ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) )
16	7, 15	syl 14	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> E. x e. RR ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) )
17	16	3adant3 1048	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( A. y e. { A , B } -. x < y /\ A. y e. RR ( y < x -> E. z e. { A , B } y < z ) ) )
18	4, 17	suplubti 7340	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C e. RR /\ C < sup ( { A , B } , RR , < ) ) -> E. z e. { A , B } C < z ) )
19	2, 18	mpand 433	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < sup ( { A , B } , RR , < ) -> E. z e. { A , B } C < z ) )
20		elpri 3732	. . . . . . . . 9 |- ( z e. { A , B } -> ( z = A \/ z = B ) )
21	20	adantr 276	. . . . . . . 8 |- ( ( z e. { A , B } /\ C < z ) -> ( z = A \/ z = B ) )
22		breq2 4134	. . . . . . . . . . 11 |- ( z = A -> ( C < z <-> C < A ) )
23	22	biimpcd 159	. . . . . . . . . 10 |- ( C < z -> ( z = A -> C < A ) )
24	23	adantl 277	. . . . . . . . 9 |- ( ( z e. { A , B } /\ C < z ) -> ( z = A -> C < A ) )
25		breq2 4134	. . . . . . . . . . 11 |- ( z = B -> ( C < z <-> C < B ) )
26	25	biimpcd 159	. . . . . . . . . 10 |- ( C < z -> ( z = B -> C < B ) )
27	26	adantl 277	. . . . . . . . 9 |- ( ( z e. { A , B } /\ C < z ) -> ( z = B -> C < B ) )
28	24, 27	orim12d 798	. . . . . . . 8 |- ( ( z e. { A , B } /\ C < z ) -> ( ( z = A \/ z = B ) -> ( C < A \/ C < B ) ) )
29	21, 28	mpd 13	. . . . . . 7 |- ( ( z e. { A , B } /\ C < z ) -> ( C < A \/ C < B ) )
30	29	rexlimiva 2663	. . . . . 6 |- ( E. z e. { A , B } C < z -> ( C < A \/ C < B ) )
31	19, 30	syl6 33	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < sup ( { A , B } , RR , < ) -> ( C < A \/ C < B ) ) )
32	31	con3d 640	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. ( C < A \/ C < B ) -> -. C < sup ( { A , B } , RR , < ) ) )
33	1, 32	biimtrrid 153	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( -. C < A /\ -. C < B ) -> -. C < sup ( { A , B } , RR , < ) ) )
34		simp1 1028	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
35	34, 2	lenltd 8444	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
36		simp2 1029	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
37	36, 2	lenltd 8444	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
38	35, 37	anbi12d 477	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ C /\ B <_ C ) <-> ( -. C < A /\ -. C < B ) ) )
39	4, 17	supclti 7338	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
40	39, 2	lenltd 8444	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) <_ C <-> -. C < sup ( { A , B } , RR , < ) ) )
41	33, 38, 40	3imtr4d 203	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ C /\ B <_ C ) -> sup ( { A , B } , RR , < ) <_ C ) )
42	41	imp 124	1 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A <_ C /\ B <_ C ) ) -> sup ( { A , B } , RR , < ) <_ C )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {cpr 3710   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   supcsup 7322   RRcr 8178   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9003   2c2 9356   abscabs 11765

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-rp 10057  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767

This theorem is used by:  maxleastb  11982  dfabsmax  11985