Theorem suplocexprlemru 7720

Description: Lemma for suplocexpr 7726. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-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
suplocexprlemru	|- ( ph -> A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )

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

Allowed substitution hints:   ph( z, u)   A( z, w, r)   B( x, y, z, u, r)

Proof of Theorem suplocexprlemru

Step	Hyp	Ref	Expression
1		suplocexpr.m	. . . . . . . . . . . 12 |- ( ph -> E. x x e. A )
2		suplocexpr.ub	. . . . . . . . . . . 12 |- ( ph -> E. x e. P. A. y e. A y <P x )
3		suplocexpr.loc	. . . . . . . . . . . 12 |- ( 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 7716	. . . . . . . . . . 11 |- ( ph -> A C_ P. )
5		suplocexpr.b	. . . . . . . . . . . 12 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
6	5	suplocexprlem2b 7715	. . . . . . . . . . 11 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
7	4, 6	syl 14	. . . . . . . . . 10 |- ( ph -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
8	7	eleq2d 2247	. . . . . . . . 9 |- ( ph -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
9	8	adantr 276	. . . . . . . 8 |- ( ( ph /\ r e. Q. ) -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
10	9	biimpa 296	. . . . . . 7 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
11		breq2 4009	. . . . . . . . 9 |- ( u = r -> ( w <Q u <-> w <Q r ) )
12	11	rexbidv 2478	. . . . . . . 8 |- ( u = r -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q r ) )
13	12	elrab 2895	. . . . . . 7 |- ( r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } <-> ( r e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q r ) )
14	10, 13	sylib 122	. . . . . 6 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> ( r e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q r ) )
15	14	simprd 114	. . . . 5 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> E. w e. |^| ( 2nd " A ) w <Q r )
16		ltbtwnnqq 7416	. . . . . . . 8 |- ( w <Q r <-> E. q e. Q. ( w <Q q /\ q <Q r ) )
17	16	biimpi 120	. . . . . . 7 |- ( w <Q r -> E. q e. Q. ( w <Q q /\ q <Q r ) )
18	17	ad2antll 491	. . . . . 6 |- ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) -> E. q e. Q. ( w <Q q /\ q <Q r ) )
19		simprr 531	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> q <Q r )
20		breq2 4009	. . . . . . . . . . . 12 |- ( u = q -> ( w <Q u <-> w <Q q ) )
21	20	rexbidv 2478	. . . . . . . . . . 11 |- ( u = q -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q q ) )
22		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> q e. Q. )
23		simprl 529	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) -> w e. |^| ( 2nd " A ) )
24	23	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> w e. |^| ( 2nd " A ) )
25		simprl 529	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> w <Q q )
26	24, 25	jca 306	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> ( w e. |^| ( 2nd " A ) /\ w <Q q ) )
27		rspe 2526	. . . . . . . . . . . 12 |- ( ( w e. |^| ( 2nd " A ) /\ w <Q q ) -> E. w e. |^| ( 2nd " A ) w <Q q )
28	26, 27	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> E. w e. |^| ( 2nd " A ) w <Q q )
29	21, 22, 28	elrabd 2897	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
30	7	eleq2d 2247	. . . . . . . . . . 11 |- ( ph -> ( q e. ( 2nd ` B ) <-> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
31	30	ad5antr 496	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> ( q e. ( 2nd ` B ) <-> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
32	29, 31	mpbird 167	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> q e. ( 2nd ` B ) )
33	19, 32	jca 306	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) /\ ( w <Q q /\ q <Q r ) ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) )
34	33	ex 115	. . . . . . 7 |- ( ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) /\ q e. Q. ) -> ( ( w <Q q /\ q <Q r ) -> ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
35	34	reximdva 2579	. . . . . 6 |- ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) -> ( E. q e. Q. ( w <Q q /\ q <Q r ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
36	18, 35	mpd 13	. . . . 5 |- ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
37	15, 36	rexlimddv 2599	. . . 4 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
38	37	ex 115	. . 3 |- ( ( ph /\ r e. Q. ) -> ( r e. ( 2nd ` B ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
39		simpllr 534	. . . . . 6 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> r e. Q. )
40		simprr 531	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> q e. ( 2nd ` B ) )
41	30	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> ( q e. ( 2nd ` B ) <-> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
42	40, 41	mpbid 147	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
43	21	elrab 2895	. . . . . . . . 9 |- ( q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } <-> ( q e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q q ) )
44	42, 43	sylib 122	. . . . . . . 8 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> ( q e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q q ) )
45	44	simprd 114	. . . . . . 7 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> E. w e. |^| ( 2nd " A ) w <Q q )
46		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> w <Q q )
47		simprl 529	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> q <Q r )
48	47	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> q <Q r )
49	46, 48	jca 306	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> ( w <Q q /\ q <Q r ) )
50		ltrelnq 7366	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
51	50	brel 4680	. . . . . . . . . . . . 13 |- ( w <Q q -> ( w e. Q. /\ q e. Q. ) )
52	51	simpld 112	. . . . . . . . . . . 12 |- ( w <Q q -> w e. Q. )
53	52	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> w e. Q. )
54		simp-4r 542	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> q e. Q. )
55	39	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> r e. Q. )
56		ltsonq 7399	. . . . . . . . . . . 12 |- <Q Or Q.
57		sotr 4320	. . . . . . . . . . . 12 |- ( ( <Q Or Q. /\ ( w e. Q. /\ q e. Q. /\ r e. Q. ) ) -> ( ( w <Q q /\ q <Q r ) -> w <Q r ) )
58	56, 57	mpan 424	. . . . . . . . . . 11 |- ( ( w e. Q. /\ q e. Q. /\ r e. Q. ) -> ( ( w <Q q /\ q <Q r ) -> w <Q r ) )
59	53, 54, 55, 58	syl3anc 1238	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> ( ( w <Q q /\ q <Q r ) -> w <Q r ) )
60	49, 59	mpd 13	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) /\ w <Q q ) -> w <Q r )
61	60	ex 115	. . . . . . . 8 |- ( ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) /\ w e. |^| ( 2nd " A ) ) -> ( w <Q q -> w <Q r ) )
62	61	reximdva 2579	. . . . . . 7 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> ( E. w e. |^| ( 2nd " A ) w <Q q -> E. w e. |^| ( 2nd " A ) w <Q r ) )
63	45, 62	mpd 13	. . . . . 6 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> E. w e. |^| ( 2nd " A ) w <Q r )
64	12, 39, 63	elrabd 2897	. . . . 5 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
65	8	ad3antrrr 492	. . . . 5 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
66	64, 65	mpbird 167	. . . 4 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> r e. ( 2nd ` B ) )
67	66	rexlimdva2 2597	. . 3 |- ( ( ph /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) )
68	38, 67	impbid 129	. 2 |- ( ( ph /\ r e. Q. ) -> ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
69	68	ralrimiva 2550	1 |- ( ph -> A. r e. Q. ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   C_ wss 3131   <.cop 3597   U.cuni 3811   |^|cint 3846   class class class wbr 4005   Or wor 4297   "cima 4631   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   <Q cltq 7286   P.cnp 7292   <P cltp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iltp 7471

This theorem is referenced by:  suplocexprlemex  7723