Theorem maxleast 11568

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 754	. . . 4 |- ( -. ( C < A \/ C < B ) <-> ( -. C < A /\ -. C < B ) )
2		simp3 1002	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		lttri3 8159	. . . . . . . . 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 11563	. . . . . . . . . . 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 985	. . . . . . . . . . 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 4050	. . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( x < y <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
9	8	notbid 669	. . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( -. x < y <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
10	9	ralbidv 2507	. . . . . . . . . . . 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 4051	. . . . . . . . . . . . . 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 2507	. . . . . . . . . . . 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 2878	. . . . . . . . . 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 1020	. . . . . . . 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 7109	. . . . . . 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 3657	. . . . . . . . 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 4051	. . . . . . . . . . 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 4051	. . . . . . . . . . 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 788	. . . . . . . 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 2619	. . . . . 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 632	. . . 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 1000	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
35	34, 2	lenltd 8197	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
36		simp2 1001	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
37	36, 2	lenltd 8197	. . . 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 7107	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
40	39, 2	lenltd 8197	. . 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 710   /\ w3a 981   = wceq 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   {cpr 3635   class class class wbr 4047   ` cfv 5276  (class class class)co 5951   supcsup 7091   RRcr 7931   + caddc 7935   < clt 8114   <_ cle 8115   - cmin 8250   / cdiv 8752   2c2 9094   abscabs 11352

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-sup 7093  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-rp 9783  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354

This theorem is referenced by:  maxleastb  11569  dfabsmax  11572