Theorem xrmaxiflemval 11480

Description: Lemma for xrmaxif 11481. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)

Hypothesis

Ref	Expression
xrmaxiflemval.m	|- M = if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )

Assertion

Ref	Expression
xrmaxiflemval	|- ( ( A e. RR* /\ B e. RR* ) -> ( M e. RR* /\ A. x e. { A , B } -. M < x /\ A. x e. RR* ( x < M -> E. z e. { A , B } x < z ) ) )

Distinct variable groups:   x, A, z   x, B, z

Allowed substitution hints:   M( x, z)

Proof of Theorem xrmaxiflemval

Step	Hyp	Ref	Expression
1		xrmaxiflemval.m	. . 3 |- M = if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) )
2		xrmaxiflemcl 11475	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) e. RR* )
3	1, 2	eqeltrid 2291	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> M e. RR* )
4		vex 2774	. . . . 5 |- x e. _V
5	4	elpr 3653	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6		xrmaxifle 11476	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
7	6, 1	breqtrrdi 4085	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ M )
8		xrlenlt 8119	. . . . . . . 8 |- ( ( A e. RR* /\ M e. RR* ) -> ( A <_ M <-> -. M < A ) )
9	3, 8	syldan 282	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ M <-> -. M < A ) )
10	7, 9	mpbid 147	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -. M < A )
11		breq2 4047	. . . . . . 7 |- ( x = A -> ( M < x <-> M < A ) )
12	11	notbid 668	. . . . . 6 |- ( x = A -> ( -. M < x <-> -. M < A ) )
13	10, 12	syl5ibrcom 157	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( x = A -> -. M < x ) )
14		xrmaxifle 11476	. . . . . . . . 9 |- ( ( B e. RR* /\ A e. RR* ) -> B <_ if ( A = +oo , +oo , if ( A = -oo , B , if ( B = +oo , +oo , if ( B = -oo , A , sup ( { B , A } , RR , < ) ) ) ) ) )
15	14	ancoms 268	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ if ( A = +oo , +oo , if ( A = -oo , B , if ( B = +oo , +oo , if ( B = -oo , A , sup ( { B , A } , RR , < ) ) ) ) ) )
16		xrmaxiflemcom 11479	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* ) -> if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) = if ( A = +oo , +oo , if ( A = -oo , B , if ( B = +oo , +oo , if ( B = -oo , A , sup ( { B , A } , RR , < ) ) ) ) ) )
17	1, 16	eqtrid 2249	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> M = if ( A = +oo , +oo , if ( A = -oo , B , if ( B = +oo , +oo , if ( B = -oo , A , sup ( { B , A } , RR , < ) ) ) ) ) )
18	15, 17	breqtrrd 4071	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ M )
19		simpr 110	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
20		xrlenlt 8119	. . . . . . . 8 |- ( ( B e. RR* /\ M e. RR* ) -> ( B <_ M <-> -. M < B ) )
21	19, 3, 20	syl2anc 411	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( B <_ M <-> -. M < B ) )
22	18, 21	mpbid 147	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* ) -> -. M < B )
23		breq2 4047	. . . . . . 7 |- ( x = B -> ( M < x <-> M < B ) )
24	23	notbid 668	. . . . . 6 |- ( x = B -> ( -. M < x <-> -. M < B ) )
25	22, 24	syl5ibrcom 157	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( x = B -> -. M < x ) )
26	13, 25	jaod 718	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( x = A \/ x = B ) -> -. M < x ) )
27	5, 26	biimtrid 152	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. { A , B } -> -. M < x ) )
28	27	ralrimiv 2577	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> A. x e. { A , B } -. M < x )
29		prid1g 3736	. . . . . . 7 |- ( A e. RR* -> A e. { A , B } )
30	29	ad4antr 494	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) /\ x < A ) -> A e. { A , B } )
31		breq2 4047	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
32	31	rspcev 2876	. . . . . 6 |- ( ( A e. { A , B } /\ x < A ) -> E. z e. { A , B } x < z )
33	30, 32	sylancom 420	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) /\ x < A ) -> E. z e. { A , B } x < z )
34		prid2g 3737	. . . . . . 7 |- ( B e. RR* -> B e. { A , B } )
35	34	ad4antlr 495	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) /\ x < B ) -> B e. { A , B } )
36		breq2 4047	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
37	36	rspcev 2876	. . . . . 6 |- ( ( B e. { A , B } /\ x < B ) -> E. z e. { A , B } x < z )
38	35, 37	sylancom 420	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) /\ x < B ) -> E. z e. { A , B } x < z )
39		simplll 533	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> A e. RR* )
40		simpllr 534	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> B e. RR* )
41		simplr 528	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> x e. RR* )
42	1	breq2i 4051	. . . . . . . 8 |- ( x < M <-> x < if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
43	42	biimpi 120	. . . . . . 7 |- ( x < M -> x < if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
44	43	adantl 277	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> x < if ( B = +oo , +oo , if ( B = -oo , A , if ( A = +oo , +oo , if ( A = -oo , B , sup ( { A , B } , RR , < ) ) ) ) ) )
45	39, 40, 41, 44	xrmaxiflemlub 11478	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> ( x < A \/ x < B ) )
46	33, 38, 45	mpjaodan 799	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> E. z e. { A , B } x < z )
47	46	ex 115	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) -> ( x < M -> E. z e. { A , B } x < z ) )
48	47	ralrimiva 2578	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> A. x e. RR* ( x < M -> E. z e. { A , B } x < z ) )
49	3, 28, 48	3jca 1179	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( M e. RR* /\ A. x e. { A , B } -. M < x /\ A. x e. RR* ( x < M -> E. z e. { A , B } x < z ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   ifcif 3570   {cpr 3633   class class class wbr 4043   supcsup 7066   RRcr 7906   +oocpnf 8086   -oocmnf 8087   RR*cxr 8088   < clt 8089   <_ cle 8090

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-frec 6467  df-sup 7068  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-rp 9758  df-seqfrec 10574  df-exp 10665  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229

This theorem is referenced by:  xrmaxif  11481