Theorem suplocexprlemex 7905

Description: Lemma for suplocexpr 7908. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	|- ( ph -> E. x x e. A )
suplocexpr.ub	|- ( ph -> E. x e. P. A. y e. A y <P x )
suplocexpr.loc	|- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )
suplocexpr.b	|- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.

Assertion

Ref	Expression
suplocexprlemex	|- ( ph -> B e. P. )

Distinct variable groups:   u, A, w, z   x, A, u, y, z   w, B   ph, u, w, z   ph, x, y

Allowed substitution hints:   B( x, y, z, u)

Proof of Theorem suplocexprlemex

Dummy variables q r s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.b	. . 3 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
2		suplocexpr.m	. . . . . 6 |- ( ph -> E. x x e. A )
3		suplocexpr.ub	. . . . . 6 |- ( ph -> E. x e. P. A. y e. A y <P x )
4		suplocexpr.loc	. . . . . 6 |- ( ph -> A. x e. P. A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) )
5	2, 3, 4	suplocexprlemss 7898	. . . . 5 |- ( ph -> A C_ P. )
6	1	suplocexprlem2b 7897	. . . . 5 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
7	5, 6	syl 14	. . . 4 |- ( ph -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
8	7	opeq2d 3863	. . 3 |- ( ph -> <. U. ( 1st " A ) , ( 2nd ` B ) >. = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
9	1, 8	eqtr4id 2281	. 2 |- ( ph -> B = <. U. ( 1st " A ) , ( 2nd ` B ) >. )
10		suplocexprlemell 7896	. . . . . . . . 9 |- ( s e. U. ( 1st " A ) <-> E. t e. A s e. ( 1st ` t ) )
11	10	biimpi 120	. . . . . . . 8 |- ( s e. U. ( 1st " A ) -> E. t e. A s e. ( 1st ` t ) )
12	11	adantl 277	. . . . . . 7 |- ( ( ph /\ s e. U. ( 1st " A ) ) -> E. t e. A s e. ( 1st ` t ) )
13	5	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> A C_ P. )
14		simprl 529	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. A )
15	13, 14	sseldd 3225	. . . . . . . . 9 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. P. )
16		prop 7658	. . . . . . . . 9 |- ( t e. P. -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
17	15, 16	syl 14	. . . . . . . 8 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. )
18		simprr 531	. . . . . . . 8 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. ( 1st ` t ) )
19		elprnql 7664	. . . . . . . 8 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ s e. ( 1st ` t ) ) -> s e. Q. )
20	17, 18, 19	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. Q. )
21	12, 20	rexlimddv 2653	. . . . . 6 |- ( ( ph /\ s e. U. ( 1st " A ) ) -> s e. Q. )
22	21	ex 115	. . . . 5 |- ( ph -> ( s e. U. ( 1st " A ) -> s e. Q. ) )
23	22	ssrdv 3230	. . . 4 |- ( ph -> U. ( 1st " A ) C_ Q. )
24		ssrab2 3309	. . . . 5 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } C_ Q.
25	7, 24	eqsstrdi 3276	. . . 4 |- ( ph -> ( 2nd ` B ) C_ Q. )
26	2, 3, 4	suplocexprlemml 7899	. . . . 5 |- ( ph -> E. q e. Q. q e. U. ( 1st " A ) )
27	2, 3, 4, 1	suplocexprlemmu 7901	. . . . 5 |- ( ph -> E. r e. Q. r e. ( 2nd ` B ) )
28	26, 27	jca 306	. . . 4 |- ( ph -> ( E. q e. Q. q e. U. ( 1st " A ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
29	23, 25, 28	jca31 309	. . 3 |- ( ph -> ( ( U. ( 1st " A ) C_ Q. /\ ( 2nd ` B ) C_ Q. ) /\ ( E. q e. Q. q e. U. ( 1st " A ) /\ E. r e. Q. r e. ( 2nd ` B ) ) ) )
30	2, 3, 4	suplocexprlemrl 7900	. . . . 5 |- ( ph -> A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )
31	2, 3, 4, 1	suplocexprlemru 7902	. . . . 5 |- ( ph -> A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
32	30, 31	jca 306	. . . 4 |- ( ph -> ( A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) )
33	2, 3, 4, 1	suplocexprlemdisj 7903	. . . 4 |- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
34	2, 3, 4, 1	suplocexprlemloc 7904	. . . 4 |- ( ph -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) )
35	32, 33, 34	3jca 1201	. . 3 |- ( ph -> ( ( A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) /\ A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) ) )
36		elinp 7657	. . 3 |- ( <. U. ( 1st " A ) , ( 2nd ` B ) >. e. P. <-> ( ( ( U. ( 1st " A ) C_ Q. /\ ( 2nd ` B ) C_ Q. ) /\ ( E. q e. Q. q e. U. ( 1st " A ) /\ E. r e. Q. r e. ( 2nd ` B ) ) ) /\ ( ( A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) ) /\ A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) ) ) )
37	29, 35, 36	sylanbrc 417	. 2 |- ( ph -> <. U. ( 1st " A ) , ( 2nd ` B ) >. e. P. )
38	9, 37	eqeltrd 2306	1 |- ( ph -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3197   <.cop 3669   U.cuni 3887   |^|cint 3922   class class class wbr 4082   "cima 4721   ` cfv 5317   1stc1st 6282   2ndc2nd 6283   Q.cnq 7463   <Q cltq 7468   P.cnp 7474   <P cltp 7478

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-iltp 7653

This theorem is referenced by:  suplocexprlemub  7906  suplocexpr  7908