Theorem maxleast 11378

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 753	. . . 4 |- ( -. ( C < A \/ C < B ) <-> ( -. C < A /\ -. C < B ) )
2		simp3 1001	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		lttri3 8106	. . . . . . . . 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 11373	. . . . . . . . . . 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 984	. . . . . . . . . . 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 4036	. . . . . . . . . . . . . 14 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( x < y <-> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
9	8	notbid 668	. . . . . . . . . . . . 13 |- ( x = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) -> ( -. x < y <-> -. ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < y ) )
10	9	ralbidv 2497	. . . . . . . . . . . 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 4037	. . . . . . . . . . . . . 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 2497	. . . . . . . . . . . 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 2868	. . . . . . . . . 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 1019	. . . . . . . 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 7066	. . . . . . 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 3645	. . . . . . . . 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 4037	. . . . . . . . . . 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 4037	. . . . . . . . . . 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 787	. . . . . . . 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 2609	. . . . . 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 999	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
35	34, 2	lenltd 8144	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ C <-> -. C < A ) )
36		simp2 1000	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
37	36, 2	lenltd 8144	. . . 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 7064	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
40	39, 2	lenltd 8144	. . 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 709   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   {cpr 3623   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   supcsup 7048   RRcr 7878   + caddc 7882   < clt 8061   <_ cle 8062   - cmin 8197   / cdiv 8699   2c2 9041   abscabs 11162

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by:  maxleastb  11379  dfabsmax  11382