Theorem suplocexprlemmu 8085

Description: Lemma for suplocexpr 8092. The upper cut of the putative supremum is inhabited. (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
suplocexprlemmu	|- ( ph -> E. s e. Q. s e. ( 2nd ` B ) )

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

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

Proof of Theorem suplocexprlemmu

Dummy variables j t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexpr.ub	. . . 4 |- ( ph -> E. x e. P. A. y e. A y <P x )
2		prop 7842	. . . . . . 7 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
3		prmu 7845	. . . . . . 7 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. -> E. s e. Q. s e. ( 2nd ` x ) )
4	2, 3	syl 14	. . . . . 6 |- ( x e. P. -> E. s e. Q. s e. ( 2nd ` x ) )
5	4	ad2antrl 494	. . . . 5 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> E. s e. Q. s e. ( 2nd ` x ) )
6		fo2nd 6392	. . . . . . . . . . . . 13 |- 2nd : _V -onto-> _V
7		fofun 5616	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 |- Fun 2nd
9		fvelima 5754	. . . . . . . . . . . 12 |- ( ( Fun 2nd /\ t e. ( 2nd " A ) ) -> E. u e. A ( 2nd ` u ) = t )
10	8, 9	mpan 428	. . . . . . . . . . 11 |- ( t e. ( 2nd " A ) -> E. u e. A ( 2nd ` u ) = t )
11	10	adantl 277	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) -> E. u e. A ( 2nd ` u ) = t )
12		suplocexpr.m	. . . . . . . . . . . . . . . 16 |- ( ph -> E. x x e. A )
13		suplocexpr.loc	. . . . . . . . . . . . . . . 16 |- ( 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 ) ) )
14	12, 1, 13	suplocexprlemss 8082	. . . . . . . . . . . . . . 15 |- ( ph -> A C_ P. )
15	14	ad5antr 500	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> A C_ P. )
16		simprl 535	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> u e. A )
17	15, 16	sseldd 3249	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> u e. P. )
18		simprl 535	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> x e. P. )
19	18	ad4antr 498	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> x e. P. )
20		breq1 4133	. . . . . . . . . . . . . . 15 |- ( y = u -> ( y <P x <-> u <P x ) )
21		simprr 537	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> A. y e. A y <P x )
22	21	ad4antr 498	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> A. y e. A y <P x )
23	20, 22, 16	rspcdva 2934	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> u <P x )
24		ltsopr 7963	. . . . . . . . . . . . . . . . 17 |- <P Or P.
25		so2nr 4466	. . . . . . . . . . . . . . . . 17 |- ( ( <P Or P. /\ ( u e. P. /\ x e. P. ) ) -> -. ( u <P x /\ x <P u ) )
26	24, 25	mpan 428	. . . . . . . . . . . . . . . 16 |- ( ( u e. P. /\ x e. P. ) -> -. ( u <P x /\ x <P u ) )
27	17, 19, 26	syl2anc 415	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> -. ( u <P x /\ x <P u ) )
28		imnan 701	. . . . . . . . . . . . . . 15 |- ( ( u <P x -> -. x <P u ) <-> -. ( u <P x /\ x <P u ) )
29	27, 28	sylibr 134	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> ( u <P x -> -. x <P u ) )
30	23, 29	mpd 13	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> -. x <P u )
31		aptiprlemu 8007	. . . . . . . . . . . . 13 |- ( ( u e. P. /\ x e. P. /\ -. x <P u ) -> ( 2nd ` x ) C_ ( 2nd ` u ) )
32	17, 19, 30, 31	syl3anc 1278	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> ( 2nd ` x ) C_ ( 2nd ` u ) )
33		simpllr 540	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> s e. ( 2nd ` x ) )
34	32, 33	sseldd 3249	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> s e. ( 2nd ` u ) )
35		simprr 537	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> ( 2nd ` u ) = t )
36	34, 35	eleqtrd 2317	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) /\ ( u e. A /\ ( 2nd ` u ) = t ) ) -> s e. t )
37	11, 36	rexlimddv 2673	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) /\ t e. ( 2nd " A ) ) -> s e. t )
38	37	ralrimiva 2623	. . . . . . . 8 |- ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) -> A. t e. ( 2nd " A ) s e. t )
39		vex 2824	. . . . . . . . 9 |- s e. _V
40	39	elint2 3977	. . . . . . . 8 |- ( s e. |^| ( 2nd " A ) <-> A. t e. ( 2nd " A ) s e. t )
41	38, 40	sylibr 134	. . . . . . 7 |- ( ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) /\ s e. ( 2nd ` x ) ) -> s e. |^| ( 2nd " A ) )
42	41	ex 115	. . . . . 6 |- ( ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) /\ s e. Q. ) -> ( s e. ( 2nd ` x ) -> s e. |^| ( 2nd " A ) ) )
43	42	reximdva 2652	. . . . 5 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> ( E. s e. Q. s e. ( 2nd ` x ) -> E. s e. Q. s e. |^| ( 2nd " A ) ) )
44	5, 43	mpd 13	. . . 4 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> E. s e. Q. s e. |^| ( 2nd " A ) )
45	1, 44	rexlimddv 2673	. . 3 |- ( ph -> E. s e. Q. s e. |^| ( 2nd " A ) )
46		simprr 537	. . . . . . 7 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> s e. |^| ( 2nd " A ) )
47		simprl 535	. . . . . . . . 9 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> s e. Q. )
48		1nq 7733	. . . . . . . . 9 |- 1Q e. Q.
49		addclnq 7742	. . . . . . . . 9 |- ( ( s e. Q. /\ 1Q e. Q. ) -> ( s +Q 1Q ) e. Q. )
50	47, 48, 49	sylancl 417	. . . . . . . 8 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> ( s +Q 1Q ) e. Q. )
51		ltaddnq 7774	. . . . . . . . 9 |- ( ( s e. Q. /\ 1Q e. Q. ) -> s <Q ( s +Q 1Q ) )
52	47, 48, 51	sylancl 417	. . . . . . . 8 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> s <Q ( s +Q 1Q ) )
53		breq2 4134	. . . . . . . . 9 |- ( j = ( s +Q 1Q ) -> ( s <Q j <-> s <Q ( s +Q 1Q ) ) )
54	53	rspcev 2929	. . . . . . . 8 |- ( ( ( s +Q 1Q ) e. Q. /\ s <Q ( s +Q 1Q ) ) -> E. j e. Q. s <Q j )
55	50, 52, 54	syl2anc 415	. . . . . . 7 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. j e. Q. s <Q j )
56		breq1 4133	. . . . . . . . 9 |- ( w = s -> ( w <Q j <-> s <Q j ) )
57	56	rexbidv 2551	. . . . . . . 8 |- ( w = s -> ( E. j e. Q. w <Q j <-> E. j e. Q. s <Q j ) )
58	57	rspcev 2929	. . . . . . 7 |- ( ( s e. |^| ( 2nd " A ) /\ E. j e. Q. s <Q j ) -> E. w e. |^| ( 2nd " A ) E. j e. Q. w <Q j )
59	46, 55, 58	syl2anc 415	. . . . . 6 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. w e. |^| ( 2nd " A ) E. j e. Q. w <Q j )
60		rexcom 2715	. . . . . 6 |- ( E. w e. |^| ( 2nd " A ) E. j e. Q. w <Q j <-> E. j e. Q. E. w e. |^| ( 2nd " A ) w <Q j )
61	59, 60	sylib 122	. . . . 5 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. j e. Q. E. w e. |^| ( 2nd " A ) w <Q j )
62		ssid 3268	. . . . . 6 |- Q. C_ Q.
63		rexss 3315	. . . . . 6 |- ( Q. C_ Q. -> ( E. j e. Q. E. w e. |^| ( 2nd " A ) w <Q j <-> E. j e. Q. ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) ) )
64	62, 63	ax-mp 5	. . . . 5 |- ( E. j e. Q. E. w e. |^| ( 2nd " A ) w <Q j <-> E. j e. Q. ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) )
65	61, 64	sylib 122	. . . 4 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. j e. Q. ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) )
66		suplocexpr.b	. . . . . . . . . 10 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
67	66	suplocexprlem2b 8081	. . . . . . . . 9 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
68	14, 67	syl 14	. . . . . . . 8 |- ( ph -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
69	68	eleq2d 2308	. . . . . . 7 |- ( ph -> ( j e. ( 2nd ` B ) <-> j e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
70		breq2 4134	. . . . . . . . 9 |- ( u = j -> ( w <Q u <-> w <Q j ) )
71	70	rexbidv 2551	. . . . . . . 8 |- ( u = j -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q j ) )
72	71	elrab 2982	. . . . . . 7 |- ( j e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } <-> ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) )
73	69, 72	bitrdi 196	. . . . . 6 |- ( ph -> ( j e. ( 2nd ` B ) <-> ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) ) )
74	73	rexbidv 2551	. . . . 5 |- ( ph -> ( E. j e. Q. j e. ( 2nd ` B ) <-> E. j e. Q. ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) ) )
75	74	adantr 276	. . . 4 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> ( E. j e. Q. j e. ( 2nd ` B ) <-> E. j e. Q. ( j e. Q. /\ E. w e. |^| ( 2nd " A ) w <Q j ) ) )
76	65, 75	mpbird 167	. . 3 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. j e. Q. j e. ( 2nd ` B ) )
77	45, 76	rexlimddv 2673	. 2 |- ( ph -> E. j e. Q. j e. ( 2nd ` B ) )
78		eleq1w 2299	. . 3 |- ( j = s -> ( j e. ( 2nd ` B ) <-> s e. ( 2nd ` B ) ) )
79	78	cbvrexv 2787	. 2 |- ( E. j e. Q. j e. ( 2nd ` B ) <-> E. s e. Q. s e. ( 2nd ` B ) )
80	77, 79	sylib 122	1 |- ( ph -> E. s e. Q. s e. ( 2nd ` B ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   <.cop 3712   U.cuni 3935   |^|cint 3970   class class class wbr 4130   Or wor 4440   "cima 4777   Fun wfun 5371   -onto->wfo 5375   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   1Qc1q 7648   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iltp 7837

This theorem is used by:  suplocexprlemex  8089