Theorem xrmaxiflemval 11873

Description: Lemma for xrmaxif 11874. 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 11868	. . 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 2318	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> M e. RR* )
4		vex 2806	. . . . 5 |- x e. _V
5	4	elpr 3694	. . . 4 |- ( x e. { A , B } <-> ( x = A \/ x = B ) )
6		xrmaxifle 11869	. . . . . . . 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 4135	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> A <_ M )
8		xrlenlt 8286	. . . . . . . 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 4097	. . . . . . 7 |- ( x = A -> ( M < x <-> M < A ) )
12	11	notbid 673	. . . . . 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 11869	. . . . . . . . 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 11872	. . . . . . . . 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 2276	. . . . . . . 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 4121	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> B <_ M )
19		simpr 110	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* )
20		xrlenlt 8286	. . . . . . . 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 4097	. . . . . . 7 |- ( x = B -> ( M < x <-> M < B ) )
24	23	notbid 673	. . . . . 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 725	. . . 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 2605	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> A. x e. { A , B } -. M < x )
29		prid1g 3779	. . . . . . 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 4097	. . . . . . 7 |- ( z = A -> ( x < z <-> x < A ) )
32	31	rspcev 2911	. . . . . 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 3780	. . . . . . 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 4097	. . . . . . 7 |- ( z = B -> ( x < z <-> x < B ) )
37	36	rspcev 2911	. . . . . 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 535	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> A e. RR* )
40		simpllr 536	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> B e. RR* )
41		simplr 529	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> x e. RR* )
42	1	breq2i 4101	. . . . . . . 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 11871	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ x e. RR* ) /\ x < M ) -> ( x < A \/ x < B ) )
46	33, 38, 45	mpjaodan 806	. . . 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 2606	. 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 1204	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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   ifcif 3607   {cpr 3674   class class class wbr 4093   supcsup 7224   RRcr 8074   +oocpnf 8253   -oocmnf 8254   RR*cxr 8255   < clt 8256   <_ cle 8257

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-rp 9933  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622

This theorem is referenced by:  xrmaxif  11874