Theorem maxleast 11745

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 757	. . . 4 |- ( -. ( C < A \/ C < B ) <-> ( -. C < A /\ -. C < B ) )
2		simp3 1023	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		lttri3 8242	. . . . . . . . 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 11740	. . . . . . . . . . 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 1006	. . . . . . . . . . 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 4086	. . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( x < y <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
9	8	notbid 671	. . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( -. x < y <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
10	9	ralbidv 2530	. . . . . . . . . . . 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 4087	. . . . . . . . . . . . . 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 2530	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 2907	. . . . . . . . . 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 1041	. . . . . . . 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 7183	. . . . . . 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 429	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C < sup ( { A , B } , RR , < ) -> E. z e. { A , B } C < z ) )
20		elpri 3689	. . . . . . . . 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 4087	. . . . . . . . . . 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 4087	. . . . . . . . . . 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 791	. . . . . . . 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 2643	. . . . . 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 634	. . . 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 1021	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
35	34, 2	lenltd 8280	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
36		simp2 1022	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
37	36, 2	lenltd 8280	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> -. C < B ) )
38	35, 37	anbi12d 473	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ C /\ B <_ C ) <-> ( -. C < A /\ -. C < B ) ) )
39	4, 17	supclti 7181	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
40	39, 2	lenltd 8280	. . 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 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {cpr 3667   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   supcsup 7165   RRcr 8014   + caddc 8018   < clt 8197   <_ cle 8198   - cmin 8333   / cdiv 8835   2c2 9177   abscabs 11529

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-sup 7167  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-rp 9867  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531

This theorem is referenced by:  maxleastb  11746  dfabsmax  11749