Theorem suplocexprlemex 7530

Description: Lemma for suplocexpr 7533. 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.m	. . . . . 6 |- ( ph -> E. x x e. A )
2		suplocexpr.ub	. . . . . 6 |- ( ph -> E. x e. P. A. y e. A y <P x )
3		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 ) ) )
4	1, 2, 3	suplocexprlemss 7523	. . . . 5 |- ( ph -> A C_ P. )
5		suplocexpr.b	. . . . . 6 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
6	5	suplocexprlem2b 7522	. . . . 5 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
7	4, 6	syl 14	. . . 4 |- ( ph -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
8	7	opeq2d 3712	. . 3 |- ( ph -> <. U. ( 1st " A ) , ( 2nd ` B ) >. = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >. )
9	8, 5	syl6reqr 2191	. 2 |- ( ph -> B = <. U. ( 1st " A ) , ( 2nd ` B ) >. )
10		suplocexprlemell 7521	. . . . . . . . 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	4	ad2antrr 479	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> A C_ P. )
14		simprl 520	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. A )
15	13, 14	sseldd 3098	. . . . . . . . 9 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. P. )
16		prop 7283	. . . . . . . . 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 521	. . . . . . . 8 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. ( 1st ` t ) )
19		elprnql 7289	. . . . . . . 8 |- ( ( <. ( 1st ` t ) , ( 2nd ` t ) >. e. P. /\ s e. ( 1st ` t ) ) -> s e. Q. )
20	17, 18, 19	syl2anc 408	. . . . . . 7 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. Q. )
21	12, 20	rexlimddv 2554	. . . . . 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 3103	. . . 4 |- ( ph -> U. ( 1st " A ) C_ Q. )
24		ssrab2 3182	. . . . 5 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } C_ Q.
25	7, 24	eqsstrdi 3149	. . . 4 |- ( ph -> ( 2nd ` B ) C_ Q. )
26	1, 2, 3	suplocexprlemml 7524	. . . . 5 |- ( ph -> E. q e. Q. q e. U. ( 1st " A ) )
27	1, 2, 3, 5	suplocexprlemmu 7526	. . . . 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	1, 2, 3	suplocexprlemrl 7525	. . . . 5 |- ( ph -> A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )
31	1, 2, 3, 5	suplocexprlemru 7527	. . . . 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	1, 2, 3, 5	suplocexprlemdisj 7528	. . . 4 |- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
34	1, 2, 3, 5	suplocexprlemloc 7529	. . . 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 1161	. . 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 7282	. . 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 413	. 2 |- ( ph -> <. U. ( 1st " A ) , ( 2nd ` B ) >. e. P. )
38	9, 37	eqeltrd 2216	1 |- ( ph -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   C_ wss 3071   <.cop 3530   U.cuni 3736   |^|cint 3771   class class class wbr 3929   "cima 4542   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   <Q cltq 7093   P.cnp 7099   <P cltp 7103

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iltp 7278

This theorem is referenced by:  suplocexprlemub  7531  suplocexpr  7533