Theorem suplocexprlemru 7674

Description: Lemma for suplocexpr 7680. 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 7670	. . . . . . . . . . 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 7669	. . . . . . . . . . 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 2240	. . . . . . . . 9 |- ( ph -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
9	8	adantr 274	. . . . . . . 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 294	. . . . . . 7 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
11		breq2 3991	. . . . . . . . 9 |- ( u = r -> ( w <Q u <-> w <Q r ) )
12	11	rexbidv 2471	. . . . . . . 8 |- ( u = r -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q r ) )
13	12	elrab 2886	. . . . . . 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 121	. . . . . 6 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> ( r e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q r ) )
15	14	simprd 113	. . . . 5 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> E. w e. |^| ( 2nd " A ) w <Q r )
16		ltbtwnnqq 7370	. . . . . . . 8 |- ( w <Q r <-> E. q e. Q. ( w <Q q /\ q <Q r ) )
17	16	biimpi 119	. . . . . . 7 |- ( w <Q r -> E. q e. Q. ( w <Q q /\ q <Q r ) )
18	17	ad2antll 488	. . . . . 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 527	. . . . . . . . 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 3991	. . . . . . . . . . . 12 |- ( u = q -> ( w <Q u <-> w <Q q ) )
21	20	rexbidv 2471	. . . . . . . . . . 11 |- ( u = q -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q q ) )
22		simplr 525	. . . . . . . . . . 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 526	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) /\ ( w e. |^| ( 2nd " A ) /\ w <Q r ) ) -> w e. |^| ( 2nd " A ) )
24	23	ad2antrr 485	. . . . . . . . . . . . 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 526	. . . . . . . . . . . . 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 304	. . . . . . . . . . . 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 2519	. . . . . . . . . . . 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 2888	. . . . . . . . . 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 2240	. . . . . . . . . . 11 |- ( ph -> ( q e. ( 2nd ` B ) <-> q e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
31	30	ad5antr 493	. . . . . . . . . 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 166	. . . . . . . . 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 304	. . . . . . . 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 114	. . . . . . 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 2572	. . . . . 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 2592	. . . 4 |- ( ( ( ph /\ r e. Q. ) /\ r e. ( 2nd ` B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) )
38	37	ex 114	. . 3 |- ( ( ph /\ r e. Q. ) -> ( r e. ( 2nd ` B ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
39		simpllr 529	. . . . . 6 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> r e. Q. )
40		simprr 527	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> q e. ( 2nd ` B ) )
41	30	ad3antrrr 489	. . . . . . . . . 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 146	. . . . . . . . 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 2886	. . . . . . . . 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 121	. . . . . . . 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 113	. . . . . . 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 109	. . . . . . . . . . 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 526	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> q <Q r )
48	47	ad2antrr 485	. . . . . . . . . . 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 304	. . . . . . . . . 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 7320	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
51	50	brel 4661	. . . . . . . . . . . . 13 |- ( w <Q q -> ( w e. Q. /\ q e. Q. ) )
52	51	simpld 111	. . . . . . . . . . . 12 |- ( w <Q q -> w e. Q. )
53	52	adantl 275	. . . . . . . . . . 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 537	. . . . . . . . . . 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 485	. . . . . . . . . . 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 7353	. . . . . . . . . . . 12 |- <Q Or Q.
57		sotr 4301	. . . . . . . . . . . 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 422	. . . . . . . . . . 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 1233	. . . . . . . . . 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 114	. . . . . . . 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 2572	. . . . . . 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 2888	. . . . 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 489	. . . . 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 166	. . . 4 |- ( ( ( ( ph /\ r e. Q. ) /\ q e. Q. ) /\ ( q <Q r /\ q e. ( 2nd ` B ) ) ) -> r e. ( 2nd ` B ) )
67	66	rexlimdva2 2590	. . 3 |- ( ( ph /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) -> r e. ( 2nd ` B ) ) )
68	38, 67	impbid 128	. 2 |- ( ( ph /\ r e. Q. ) -> ( r e. ( 2nd ` B ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` B ) ) ) )
69	68	ralrimiva 2543	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 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   C_ wss 3121   <.cop 3584   U.cuni 3794   |^|cint 3829   class class class wbr 3987   Or wor 4278   "cima 4612   ` cfv 5196   1stc1st 6115   2ndc2nd 6116   Q.cnq 7235   <Q cltq 7240   P.cnp 7246   <P cltp 7250

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-1o 6393  df-oadd 6397  df-omul 6398  df-er 6511  df-ec 6513  df-qs 6517  df-ni 7259  df-pli 7260  df-mi 7261  df-lti 7262  df-plpq 7299  df-mpq 7300  df-enq 7302  df-nqqs 7303  df-plqqs 7304  df-mqqs 7305  df-1nqqs 7306  df-rq 7307  df-ltnqqs 7308  df-inp 7421  df-iltp 7425

This theorem is referenced by:  suplocexprlemex  7677