Theorem suplocexprlemrl 8034

Description: Lemma for suplocexpr 8042. The lower 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 ) ) )

Assertion

Ref	Expression
suplocexprlemrl	|- ( ph -> A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )

Distinct variable groups:   A, r   x, A, y   ph, q, r   ph, x, y

Allowed substitution hints:   ph( z)   A( z, q)

Proof of Theorem suplocexprlemrl

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suplocexprlemell 8030	. . . . . . 7 |- ( q e. U. ( 1st " A ) <-> E. s e. A q e. ( 1st ` s ) )
2	1	biimpi 120	. . . . . 6 |- ( q e. U. ( 1st " A ) -> E. s e. A q e. ( 1st ` s ) )
3	2	adantl 277	. . . . 5 |- ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) -> E. s e. A q e. ( 1st ` s ) )
4		suplocexpr.m	. . . . . . . . . . 11 |- ( ph -> E. x x e. A )
5		suplocexpr.ub	. . . . . . . . . . 11 |- ( ph -> E. x e. P. A. y e. A y <P x )
6		suplocexpr.loc	. . . . . . . . . . 11 |- ( 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 ) ) )
7	4, 5, 6	suplocexprlemss 8032	. . . . . . . . . 10 |- ( ph -> A C_ P. )
8	7	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> A C_ P. )
9		simprl 531	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> s e. A )
10	8, 9	sseldd 3241	. . . . . . . 8 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> s e. P. )
11		prop 7792	. . . . . . . 8 |- ( s e. P. -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
12	10, 11	syl 14	. . . . . . 7 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
13		simprr 533	. . . . . . 7 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> q e. ( 1st ` s ) )
14		prnmaxl 7805	. . . . . . 7 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ q e. ( 1st ` s ) ) -> E. r e. ( 1st ` s ) q <Q r )
15	12, 13, 14	syl2anc 411	. . . . . 6 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> E. r e. ( 1st ` s ) q <Q r )
16		ltrelnq 7682	. . . . . . . . 9 |- <Q C_ ( Q. X. Q. )
17	16	brel 4804	. . . . . . . 8 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
18	17	simprd 114	. . . . . . 7 |- ( q <Q r -> r e. Q. )
19	18	ad2antll 491	. . . . . 6 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> r e. Q. )
20		simprr 533	. . . . . . 7 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> q <Q r )
21		simplrl 537	. . . . . . . . 9 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> s e. A )
22		simprl 531	. . . . . . . . 9 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> r e. ( 1st ` s ) )
23		rspe 2593	. . . . . . . . 9 |- ( ( s e. A /\ r e. ( 1st ` s ) ) -> E. s e. A r e. ( 1st ` s ) )
24	21, 22, 23	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> E. s e. A r e. ( 1st ` s ) )
25		suplocexprlemell 8030	. . . . . . . 8 |- ( r e. U. ( 1st " A ) <-> E. s e. A r e. ( 1st ` s ) )
26	24, 25	sylibr 134	. . . . . . 7 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> r e. U. ( 1st " A ) )
27	20, 26	jca 306	. . . . . 6 |- ( ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) /\ ( r e. ( 1st ` s ) /\ q <Q r ) ) -> ( q <Q r /\ r e. U. ( 1st " A ) ) )
28	15, 19, 27	reximssdv 2648	. . . . 5 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) )
29	3, 28	rexlimddv 2667	. . . 4 |- ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) -> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) )
30	29	ex 115	. . 3 |- ( ( ph /\ q e. Q. ) -> ( q e. U. ( 1st " A ) -> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )
31		simprr 533	. . . . . . 7 |- ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) -> r e. U. ( 1st " A ) )
32	31, 25	sylib 122	. . . . . 6 |- ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) -> E. s e. A r e. ( 1st ` s ) )
33		simprl 531	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> s e. A )
34		simplrl 537	. . . . . . . . . 10 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> q <Q r )
35	7	ad3antrrr 492	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> A C_ P. )
36	35, 33	sseldd 3241	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> s e. P. )
37	36, 11	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. )
38		simprr 533	. . . . . . . . . . 11 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> r e. ( 1st ` s ) )
39		prcdnql 7801	. . . . . . . . . . 11 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ r e. ( 1st ` s ) ) -> ( q <Q r -> q e. ( 1st ` s ) ) )
40	37, 38, 39	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> ( q <Q r -> q e. ( 1st ` s ) ) )
41	34, 40	mpd 13	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> q e. ( 1st ` s ) )
42		19.8a 1639	. . . . . . . . 9 |- ( ( s e. A /\ q e. ( 1st ` s ) ) -> E. s ( s e. A /\ q e. ( 1st ` s ) ) )
43	33, 41, 42	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> E. s ( s e. A /\ q e. ( 1st ` s ) ) )
44		df-rex 2528	. . . . . . . 8 |- ( E. s e. A q e. ( 1st ` s ) <-> E. s ( s e. A /\ q e. ( 1st ` s ) ) )
45	43, 44	sylibr 134	. . . . . . 7 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> E. s e. A q e. ( 1st ` s ) )
46	45, 1	sylibr 134	. . . . . 6 |- ( ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) /\ ( s e. A /\ r e. ( 1st ` s ) ) ) -> q e. U. ( 1st " A ) )
47	32, 46	rexlimddv 2667	. . . . 5 |- ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) -> q e. U. ( 1st " A ) )
48	47	ex 115	. . . 4 |- ( ( ph /\ q e. Q. ) -> ( ( q <Q r /\ r e. U. ( 1st " A ) ) -> q e. U. ( 1st " A ) ) )
49	48	rexlimdvw 2666	. . 3 |- ( ( ph /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) -> q e. U. ( 1st " A ) ) )
50	30, 49	impbid 129	. 2 |- ( ( ph /\ q e. Q. ) -> ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )
51	50	ralrimiva 2617	1 |- ( ph -> A. q e. Q. ( q e. U. ( 1st " A ) <-> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   C_ wss 3213   <.cop 3694   U.cuni 3916   class class class wbr 4111   "cima 4754   ` cfv 5354   1stc1st 6334   2ndc2nd 6335   Q.cnq 7597   <Q cltq 7602   P.cnp 7608   <P cltp 7612

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1st 6336  df-2nd 6337  df-qs 6775  df-ni 7621  df-nqqs 7665  df-ltnqqs 7670  df-inp 7783  df-iltp 7787

This theorem is referenced by:  suplocexprlemex  8039