Theorem suplocexprlemrl 7679

Description: Lemma for suplocexpr 7687. 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 7675	. . . . . . 7 |- ( q e. U. ( 1st " A ) <-> E. s e. A q e. ( 1st ` s ) )
2	1	biimpi 119	. . . . . 6 |- ( q e. U. ( 1st " A ) -> E. s e. A q e. ( 1st ` s ) )
3	2	adantl 275	. . . . 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 7677	. . . . . . . . . 10 |- ( ph -> A C_ P. )
8	7	ad3antrrr 489	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> A C_ P. )
9		simprl 526	. . . . . . . . 9 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> s e. A )
10	8, 9	sseldd 3148	. . . . . . . 8 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> s e. P. )
11		prop 7437	. . . . . . . 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 527	. . . . . . 7 |- ( ( ( ( ph /\ q e. Q. ) /\ q e. U. ( 1st " A ) ) /\ ( s e. A /\ q e. ( 1st ` s ) ) ) -> q e. ( 1st ` s ) )
14		prnmaxl 7450	. . . . . . 7 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ q e. ( 1st ` s ) ) -> E. r e. ( 1st ` s ) q <Q r )
15	12, 13, 14	syl2anc 409	. . . . . 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 7327	. . . . . . . . 9 |- <Q C_ ( Q. X. Q. )
17	16	brel 4663	. . . . . . . 8 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
18	17	simprd 113	. . . . . . 7 |- ( q <Q r -> r e. Q. )
19	18	ad2antll 488	. . . . . 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 527	. . . . . . 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 530	. . . . . . . . 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 526	. . . . . . . . 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 2519	. . . . . . . . 9 |- ( ( s e. A /\ r e. ( 1st ` s ) ) -> E. s e. A r e. ( 1st ` s ) )
24	21, 22, 23	syl2anc 409	. . . . . . . 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 7675	. . . . . . . 8 |- ( r e. U. ( 1st " A ) <-> E. s e. A r e. ( 1st ` s ) )
26	24, 25	sylibr 133	. . . . . . 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 304	. . . . . 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 2574	. . . . 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 2592	. . . 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 114	. . 3 |- ( ( ph /\ q e. Q. ) -> ( q e. U. ( 1st " A ) -> E. r e. Q. ( q <Q r /\ r e. U. ( 1st " A ) ) ) )
31		simprr 527	. . . . . . 7 |- ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) -> r e. U. ( 1st " A ) )
32	31, 25	sylib 121	. . . . . 6 |- ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) -> E. s e. A r e. ( 1st ` s ) )
33		simprl 526	. . . . . . . . 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 530	. . . . . . . . . 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 489	. . . . . . . . . . . . 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 3148	. . . . . . . . . . . 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 527	. . . . . . . . . . 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 7446	. . . . . . . . . . 11 |- ( ( <. ( 1st ` s ) , ( 2nd ` s ) >. e. P. /\ r e. ( 1st ` s ) ) -> ( q <Q r -> q e. ( 1st ` s ) ) )
40	37, 38, 39	syl2anc 409	. . . . . . . . . 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 1583	. . . . . . . . 9 |- ( ( s e. A /\ q e. ( 1st ` s ) ) -> E. s ( s e. A /\ q e. ( 1st ` s ) ) )
43	33, 41, 42	syl2anc 409	. . . . . . . 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 2454	. . . . . . . 8 |- ( E. s e. A q e. ( 1st ` s ) <-> E. s ( s e. A /\ q e. ( 1st ` s ) ) )
45	43, 44	sylibr 133	. . . . . . 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 133	. . . . . 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 2592	. . . . 5 |- ( ( ( ph /\ q e. Q. ) /\ ( q <Q r /\ r e. U. ( 1st " A ) ) ) -> q e. U. ( 1st " A ) )
48	47	ex 114	. . . 4 |- ( ( ph /\ q e. Q. ) -> ( ( q <Q r /\ r e. U. ( 1st " A ) ) -> q e. U. ( 1st " A ) ) )
49	48	rexlimdvw 2591	. . 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 128	. 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 2543	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 103   <-> wb 104   \/ wo 703   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   C_ wss 3121   <.cop 3586   U.cuni 3796   class class class wbr 3989   "cima 4614   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   <Q cltq 7247   P.cnp 7253   <P cltp 7257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-ltnqqs 7315  df-inp 7428  df-iltp 7432

This theorem is referenced by:  suplocexprlemex  7684