Theorem suplocexprlemru 7832

Description: Lemma for suplocexpr 7838. 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 7828	. . . . . . . . . . 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 7827	. . . . . . . . . . 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 2275	. . . . . . . . 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 4048	. . . . . . . . 9 |- ( u = r -> ( w <Q u <-> w <Q r ) )
12	11	rexbidv 2507	. . . . . . . 8 |- ( u = r -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q r ) )
13	12	elrab 2929	. . . . . . 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 7528	. . . . . . . 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 4048	. . . . . . . . . . . 12 |- ( u = q -> ( w <Q u <-> w <Q q ) )
21	20	rexbidv 2507	. . . . . . . . . . 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 2555	. . . . . . . . . . . 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 2931	. . . . . . . . . 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 2275	. . . . . . . . . . 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 2608	. . . . . 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 2628	. . . 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 2929	. . . . . . . . 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 7478	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
51	50	brel 4727	. . . . . . . . . . . . 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 7511	. . . . . . . . . . . 12 |- <Q Or Q.
57		sotr 4365	. . . . . . . . . . . 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 1250	. . . . . . . . . 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 2608	. . . . . . 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 2931	. . . . 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 2626	. . 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 2579	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 710   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   C_ wss 3166   <.cop 3636   U.cuni 3850   |^|cint 3885   class class class wbr 4044   Or wor 4342   "cima 4678   ` cfv 5271   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   <Q cltq 7398   P.cnp 7404   <P cltp 7408

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-inp 7579  df-iltp 7583

This theorem is referenced by:  suplocexprlemex  7835