Theorem maxleim 11890

Description: Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)

Assertion

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

Proof of Theorem maxleim

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

Step	Hyp	Ref	Expression
1		lttri3 8353	. . . 4 |- ( ( f e. RR /\ g e. RR ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 277	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ ( f e. RR /\ g e. RR ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		simplr 529	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) -> B e. RR )
4		prid2g 3796	. . . 4 |- ( B e. RR -> B e. { A , B } )
5	3, 4	syl 14	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) -> B e. { A , B } )
6		simpll 527	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) -> A e. RR )
7	6	ad2antrr 488	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A e. RR )
8	3	ad2antrr 488	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> B e. RR )
9		simpllr 536	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A <_ B )
10	7, 8, 9	lensymd 8395	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < A )
11		breq2 4113	. . . . . . 7 |- ( y = A -> ( B < y <-> B < A ) )
12	11	notbid 673	. . . . . 6 |- ( y = A -> ( -. B < y <-> -. B < A ) )
13	12	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( -. B < y <-> -. B < A ) )
14	10, 13	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < y )
15	3	ad2antrr 488	. . . . . 6 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> B e. RR )
16	15	ltnrd 8385	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < B )
17		breq2 4113	. . . . . . 7 |- ( y = B -> ( B < y <-> B < B ) )
18	17	notbid 673	. . . . . 6 |- ( y = B -> ( -. B < y <-> -. B < B ) )
19	18	adantl 277	. . . . 5 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> ( -. B < y <-> -. B < B ) )
20	16, 19	mpbird 167	. . . 4 |- ( ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < y )
21		elpri 3712	. . . . 5 |- ( y e. { A , B } -> ( y = A \/ y = B ) )
22	21	adantl 277	. . . 4 |- ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) -> ( y = A \/ y = B ) )
23	14, 20, 22	mpjaodan 806	. . 3 |- ( ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) /\ y e. { A , B } ) -> -. B < y )
24	2, 3, 5, 23	supmaxti 7295	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ A <_ B ) -> sup ( { A , B } , RR , < ) = B )
25	24	ex 115	1 |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> sup ( { A , B } , RR , < ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2203   {cpr 3690   class class class wbr 4109   supcsup 7273   RRcr 8126   < clt 8308   <_ cle 8309

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239  ax-pre-apti 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-iota 5312  df-riota 6003  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314

This theorem is referenced by:  maxleb  11901  xrmaxiflemab  11932