Theorem suplocexprlemru 7779

Description: Lemma for suplocexpr 7785. 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 7775	. . . . . . . . . . 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 7774	. . . . . . . . . . 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 2263	. . . . . . . . 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 4033	. . . . . . . . 9 |- ( u = r -> ( w <Q u <-> w <Q r ) )
12	11	rexbidv 2495	. . . . . . . 8 |- ( u = r -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q r ) )
13	12	elrab 2916	. . . . . . 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 7475	. . . . . . . 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 4033	. . . . . . . . . . . 12 |- ( u = q -> ( w <Q u <-> w <Q q ) )
21	20	rexbidv 2495	. . . . . . . . . . 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 2543	. . . . . . . . . . . 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 2918	. . . . . . . . . 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 2263	. . . . . . . . . . 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 2596	. . . . . 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 2616	. . . 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 2916	. . . . . . . . 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 7425	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
51	50	brel 4711	. . . . . . . . . . . . 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 7458	. . . . . . . . . . . 12 |- <Q Or Q.
57		sotr 4349	. . . . . . . . . . . 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 1249	. . . . . . . . . 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 2596	. . . . . . 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 2918	. . . . 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 2614	. . 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 2567	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 709   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   C_ wss 3153   <.cop 3621   U.cuni 3835   |^|cint 3870   class class class wbr 4029   Or wor 4326   "cima 4662   ` cfv 5254   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   <Q cltq 7345   P.cnp 7351   <P cltp 7355

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-iltp 7530

This theorem is referenced by:  suplocexprlemex  7782