Theorem xrmaxiflemval 11801

Description: Lemma for xrmaxif 11802. 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 11796	. . 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 2316	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> M e. RR* )
4		vex 2803	. . . . 5 |- x e. _V
5	4	elpr 3688	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6		xrmaxifle 11797	. . . . . . . 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 4128	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ M )
8		xrlenlt 8234	. . . . . . . 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 4090	. . . . . . 7 |- ( x = A -> ( M < x <-> M < A ) )
12	11	notbid 671	. . . . . 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 11797	. . . . . . . . 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 11800	. . . . . . . . 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 2274	. . . . . . . 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 4114	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ M )
19		simpr 110	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
20		xrlenlt 8234	. . . . . . . 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 4090	. . . . . . 7 |- ( x = B -> ( M < x <-> M < B ) )
24	23	notbid 671	. . . . . 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 722	. . . 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 2602	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> A. x e. { A , B } -. M < x )
29		prid1g 3773	. . . . . . 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 4090	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
32	31	rspcev 2908	. . . . . 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 3774	. . . . . . 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 4090	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
37	36	rspcev 2908	. . . . . 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 4094	. . . . . . . 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 11799	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> ( x < A \/ x < B ) )
46	33, 38, 45	mpjaodan 803	. . . 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 2603	. 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 1201	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 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ifcif 3603   {cpr 3668   class class class wbr 4086   supcsup 7172   RRcr 8021   +oocpnf 8201   -oocmnf 8202   RR*cxr 8203   < clt 8204   <_ cle 8205

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7174  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-rp 9879  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550

This theorem is referenced by:  xrmaxif  11802