Theorem xrmaxiflemval 11393

Description: Lemma for xrmaxif 11394. 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 11388	. . 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 2280	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> M e. RR* )
4		vex 2763	. . . . 5 |- x e. _V
5	4	elpr 3639	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6		xrmaxifle 11389	. . . . . . . 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 4071	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ M )
8		xrlenlt 8084	. . . . . . . 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 4033	. . . . . . 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 11389	. . . . . . . . 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 11392	. . . . . . . . 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 2238	. . . . . . . 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 4057	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ M )
19		simpr 110	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
20		xrlenlt 8084	. . . . . . . 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 4033	. . . . . . 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 2566	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> A. x e. { A , B } -. M < x )
29		prid1g 3722	. . . . . . 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 4033	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
32	31	rspcev 2864	. . . . . 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 3723	. . . . . . 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 4033	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
37	36	rspcev 2864	. . . . . 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 4037	. . . . . . . 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 11391	. . . . 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 2567	. 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 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   ifcif 3557   {cpr 3619   class class class wbr 4029   supcsup 7041   RRcr 7871   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   < clt 8054   <_ cle 8055

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-rp 9720  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143

This theorem is referenced by:  xrmaxif  11394