Theorem suplocexprlemmu 7861

Description: Lemma for suplocexpr 7868. 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 7618	. . . . . . 7 |- ( x e. P. -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. P. )
3		prmu 7621	. . . . . . 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 490	. . . . 5 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> E. s e. Q. s e. ( 2nd ` x ) )
6		fo2nd 6262	. . . . . . . . . . . . 13 |- 2nd : _V -onto-> _V
7		fofun 5516	. . . . . . . . . . . . 13 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 |- Fun 2nd
9		fvelima 5648	. . . . . . . . . . . 12 |- ( ( Fun 2nd /\ t e. ( 2nd " A ) ) -> E. u e. A ( 2nd ` u ) = t )
10	8, 9	mpan 424	. . . . . . . . . . 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 7858	. . . . . . . . . . . . . . 15 |- ( ph -> A C_ P. )
15	14	ad5antr 496	. . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . 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 529	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> x e. P. )
19	18	ad4antr 494	. . . . . . . . . . . . 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 4057	. . . . . . . . . . . . . . 15 |- ( y = u -> ( y <P x <-> u <P x ) )
21		simprr 531	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( x e. P. /\ A. y e. A y <P x ) ) -> A. y e. A y <P x )
22	21	ad4antr 494	. . . . . . . . . . . . . . 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 2886	. . . . . . . . . . . . . 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 7739	. . . . . . . . . . . . . . . . 17 |- <P Or P.
25		so2nr 4381	. . . . . . . . . . . . . . . . 17 |- ( ( <P Or P. /\ ( u e. P. /\ x e. P. ) ) -> -. ( u <P x /\ x <P u ) )
26	24, 25	mpan 424	. . . . . . . . . . . . . . . 16 |- ( ( u e. P. /\ x e. P. ) -> -. ( u <P x /\ x <P u ) )
27	17, 19, 26	syl2anc 411	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . 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 7783	. . . . . . . . . . . . 13 |- ( ( u e. P. /\ x e. P. /\ -. x <P u ) -> ( 2nd ` x ) C_ ( 2nd ` u ) )
32	17, 19, 30, 31	syl3anc 1250	. . . . . . . . . . . 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 534	. . . . . . . . . . . 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 3198	. . . . . . . . . . 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 531	. . . . . . . . . . 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 2285	. . . . . . . . . 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 2629	. . . . . . . . 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 2580	. . . . . . . 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 2776	. . . . . . . . 9 |- s e. _V
40	39	elint2 3901	. . . . . . . 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 2609	. . . . 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 2629	. . 3 |- ( ph -> E. s e. Q. s e. |^| ( 2nd " A ) )
46		simprr 531	. . . . . . 7 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> s e. |^| ( 2nd " A ) )
47		simprl 529	. . . . . . . . 9 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> s e. Q. )
48		1nq 7509	. . . . . . . . 9 |- 1Q e. Q.
49		addclnq 7518	. . . . . . . . 9 |- ( ( s e. Q. /\ 1Q e. Q. ) -> ( s +Q 1Q ) e. Q. )
50	47, 48, 49	sylancl 413	. . . . . . . 8 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> ( s +Q 1Q ) e. Q. )
51		ltaddnq 7550	. . . . . . . . 9 |- ( ( s e. Q. /\ 1Q e. Q. ) -> s <Q ( s +Q 1Q ) )
52	47, 48, 51	sylancl 413	. . . . . . . 8 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> s <Q ( s +Q 1Q ) )
53		breq2 4058	. . . . . . . . 9 |- ( j = ( s +Q 1Q ) -> ( s <Q j <-> s <Q ( s +Q 1Q ) ) )
54	53	rspcev 2881	. . . . . . . 8 |- ( ( ( s +Q 1Q ) e. Q. /\ s <Q ( s +Q 1Q ) ) -> E. j e. Q. s <Q j )
55	50, 52, 54	syl2anc 411	. . . . . . 7 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. j e. Q. s <Q j )
56		breq1 4057	. . . . . . . . 9 |- ( w = s -> ( w <Q j <-> s <Q j ) )
57	56	rexbidv 2508	. . . . . . . 8 |- ( w = s -> ( E. j e. Q. w <Q j <-> E. j e. Q. s <Q j ) )
58	57	rspcev 2881	. . . . . . 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 411	. . . . . 6 |- ( ( ph /\ ( s e. Q. /\ s e. |^| ( 2nd " A ) ) ) -> E. w e. |^| ( 2nd " A ) E. j e. Q. w <Q j )
60		rexcom 2671	. . . . . 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 3217	. . . . . 6 |- Q. C_ Q.
63		rexss 3264	. . . . . 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 7857	. . . . . . . . 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 2276	. . . . . . 7 |- ( ph -> ( j e. ( 2nd ` B ) <-> j e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
70		breq2 4058	. . . . . . . . 9 |- ( u = j -> ( w <Q u <-> w <Q j ) )
71	70	rexbidv 2508	. . . . . . . 8 |- ( u = j -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q j ) )
72	71	elrab 2933	. . . . . . 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 2508	. . . . 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 2629	. 2 |- ( ph -> E. j e. Q. j e. ( 2nd ` B ) )
78		eleq1w 2267	. . 3 |- ( j = s -> ( j e. ( 2nd ` B ) <-> s e. ( 2nd ` B ) ) )
79	78	cbvrexv 2740	. 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 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   E.wex 1516   e. wcel 2177   A.wral 2485   E.wrex 2486   {crab 2489   _Vcvv 2773   C_ wss 3170   <.cop 3641   U.cuni 3859   |^|cint 3894   class class class wbr 4054   Or wor 4355   "cima 4691   Fun wfun 5279   -onto->wfo 5283   ` cfv 5285  (class class class)co 5962   1stc1st 6242   2ndc2nd 6243   Q.cnq 7423   1Qc1q 7424   +Q cplq 7425   <Q cltq 7428   P.cnp 7434   <P cltp 7438

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-eprel 4349  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-1o 6520  df-2o 6521  df-oadd 6524  df-omul 6525  df-er 6638  df-ec 6640  df-qs 6644  df-ni 7447  df-pli 7448  df-mi 7449  df-lti 7450  df-plpq 7487  df-mpq 7488  df-enq 7490  df-nqqs 7491  df-plqqs 7492  df-mqqs 7493  df-1nqqs 7494  df-rq 7495  df-ltnqqs 7496  df-enq0 7567  df-nq0 7568  df-0nq0 7569  df-plq0 7570  df-mq0 7571  df-inp 7609  df-iltp 7613

This theorem is referenced by:  suplocexprlemex  7865