Theorem suplocexprlemex 7636

Description: Lemma for suplocexpr 7639. 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 7629	. . . . 5 |- ( ph -> A C_ P. )
6	1	suplocexprlem2b 7628	. . . . 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 3748	. . 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 2209	. 2 |- ( ph -> B = <. U. ( 1st " A ) , ( 2nd ` B ) >. )
10		suplocexprlemell 7627	. . . . . . . . 9 |- ( s e. U. ( 1st " A ) <-> E. t e. A s e. ( 1st ` t ) )
11	10	biimpi 119	. . . . . . . 8 |- ( s e. U. ( 1st " A ) -> E. t e. A s e. ( 1st ` t ) )
12	11	adantl 275	. . . . . . 7 |- ( ( ph /\ s e. U. ( 1st " A ) ) -> E. t e. A s e. ( 1st ` t ) )
13	5	ad2antrr 480	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> A C_ P. )
14		simprl 521	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. A )
15	13, 14	sseldd 3129	. . . . . . . . 9 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. P. )
16		prop 7389	. . . . . . . . 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 522	. . . . . . . 8 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. ( 1st ` t ) )
19		elprnql 7395	. . . . . . . 8 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ s e. ( 1st ` t ) ) -> s e. Q. )
20	17, 18, 19	syl2anc 409	. . . . . . 7 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. Q. )
21	12, 20	rexlimddv 2579	. . . . . 6 |- ( ( ph /\ s e. U. ( 1st " A ) ) -> s e. Q. )
22	21	ex 114	. . . . 5 |- ( ph -> ( s e. U. ( 1st " A ) -> s e. Q. ) )
23	22	ssrdv 3134	. . . 4 |- ( ph -> U. ( 1st " A ) C_ Q. )
24		ssrab2 3213	. . . . 5 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } C_ Q.
25	7, 24	eqsstrdi 3180	. . . 4 |- ( ph -> ( 2nd ` B ) C_ Q. )
26	2, 3, 4	suplocexprlemml 7630	. . . . 5 |- ( ph -> E. q e. Q. q e. U. ( 1st " A ) )
27	2, 3, 4, 1	suplocexprlemmu 7632	. . . . 5 |- ( ph -> E. r e. Q. r e. ( 2nd ` B ) )
28	26, 27	jca 304	. . . 4 |- ( ph -> ( E. q e. Q. q e. U. ( 1st " A ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )
29	23, 25, 28	jca31 307	. . 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 7631	. . . . 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 7633	. . . . 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 304	. . . 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 7634	. . . 4 |- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
34	2, 3, 4, 1	suplocexprlemloc 7635	. . . 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 1162	. . 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 7388	. . 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 414	. 2 |- ( ph -> <. U. ( 1st " A ) , ( 2nd ` B ) >. e. P. )
38	9, 37	eqeltrd 2234	1 |- ( ph -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 963   = wceq 1335   E.wex 1472   e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439   C_ wss 3102   <.cop 3563   U.cuni 3772   |^|cint 3807   class class class wbr 3965   "cima 4588   ` cfv 5169   1stc1st 6083   2ndc2nd 6084   Q.cnq 7194   <Q cltq 7199   P.cnp 7205   <P cltp 7209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-1o 6360  df-2o 6361  df-oadd 6364  df-omul 6365  df-er 6477  df-ec 6479  df-qs 6483  df-ni 7218  df-pli 7219  df-mi 7220  df-lti 7221  df-plpq 7258  df-mpq 7259  df-enq 7261  df-nqqs 7262  df-plqqs 7263  df-mqqs 7264  df-1nqqs 7265  df-rq 7266  df-ltnqqs 7267  df-enq0 7338  df-nq0 7339  df-0nq0 7340  df-plq0 7341  df-mq0 7342  df-inp 7380  df-iltp 7384

This theorem is referenced by:  suplocexprlemub  7637  suplocexpr  7639