Theorem xrmaxiflemval 11271

Description: Lemma for xrmaxif 11272. 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 11266	. . 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 2274	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> M e. RR* )
4		vex 2752	. . . . 5 |- x e. _V
5	4	elpr 3625	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6		xrmaxifle 11267	. . . . . . . 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 4057	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ M )
8		xrlenlt 8035	. . . . . . . 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 4019	. . . . . . 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 11267	. . . . . . . . 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 11270	. . . . . . . . 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 2232	. . . . . . . 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 4043	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ M )
19		simpr 110	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
20		xrlenlt 8035	. . . . . . . 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 4019	. . . . . . 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 2559	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> A. x e. { A , B } -. M < x )
29		prid1g 3708	. . . . . . 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 4019	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
32	31	rspcev 2853	. . . . . 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 3709	. . . . . . 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 4019	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
37	36	rspcev 2853	. . . . . 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 4023	. . . . . . . 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 11269	. . . . 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 2560	. 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 1178	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 979   = wceq 1363   e. wcel 2158   A.wral 2465   E.wrex 2466   ifcif 3546   {cpr 3605   class class class wbr 4015   supcsup 6994   RRcr 7823   +oocpnf 8002   -oocmnf 8003   RR*cxr 8004   < clt 8005   <_ cle 8006

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021

This theorem is referenced by:  xrmaxif  11272