Theorem suplocexprlemex 7941

Description: Lemma for suplocexpr 7944. 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 7934	. . . . 5 |- ( ph -> A C_ P. )
6	1	suplocexprlem2b 7933	. . . . 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 3869	. . 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 2283	. 2 |- ( ph -> B = <. U. ( 1st " A ) , ( 2nd ` B ) >. )
10		suplocexprlemell 7932	. . . . . . . . 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 531	. . . . . . . . . 10 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. A )
15	13, 14	sseldd 3228	. . . . . . . . 9 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> t e. P. )
16		prop 7694	. . . . . . . . 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 533	. . . . . . . 8 |- ( ( ( ph /\ s e. U. ( 1st " A ) ) /\ ( t e. A /\ s e. ( 1st ` t ) ) ) -> s e. ( 1st ` t ) )
19		elprnql 7700	. . . . . . . 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 2655	. . . . . 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 3233	. . . 4 |- ( ph -> U. ( 1st " A ) C_ Q. )
24		ssrab2 3312	. . . . 5 |- { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } C_ Q.
25	7, 24	eqsstrdi 3279	. . . 4 |- ( ph -> ( 2nd ` B ) C_ Q. )
26	2, 3, 4	suplocexprlemml 7935	. . . . 5 |- ( ph -> E. q e. Q. q e. U. ( 1st " A ) )
27	2, 3, 4, 1	suplocexprlemmu 7937	. . . . 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 7936	. . . . 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 7938	. . . . 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 7939	. . . 4 |- ( ph -> A. q e. Q. -. ( q e. U. ( 1st " A ) /\ q e. ( 2nd ` B ) ) )
34	2, 3, 4, 1	suplocexprlemloc 7940	. . . 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 1203	. . 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 7693	. . 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 2308	1 |- ( ph -> B e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   {crab 2514   C_ wss 3200   <.cop 3672   U.cuni 3893   |^|cint 3928   class class class wbr 4088   "cima 4728   ` cfv 5326   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   <Q cltq 7504   P.cnp 7510   <P cltp 7514

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-iltp 7689

This theorem is referenced by:  suplocexprlemub  7942  suplocexpr  7944