Theorem suplocsrlemb 8137

Description: Lemma for suplocsr 8140. The set B is located. (Contributed by Jim Kingdon, 18-Jan-2024.)

Hypotheses

Ref	Expression
suplocsrlem.b	|- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }
suplocsrlem.ss	|- ( ph -> A C_ R. )
suplocsrlem.c	|- ( ph -> C e. A )
suplocsrlem.ub	|- ( ph -> E. x e. R. A. y e. A y <R x )
suplocsrlem.loc	|- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )

Assertion

Ref	Expression
suplocsrlemb	|- ( ph -> A. u e. P. A. v e. P. ( u <P v -> ( E. q e. B u <P q \/ A. q e. B q <P v ) ) )

Distinct variable groups:   A, q, w   x, A, y, z   z, B   C, q, w   x, C, y, z   ph, q, u, v, z   x, u, y   y, v

Allowed substitution hints:   ph( x, y, w)   A( v, u)   B( x, y, w, v, u, q)   C( v, u)

Proof of Theorem suplocsrlemb

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> u <P v )
2		simplrl 537	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> u e. P. )
3		simplrr 538	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> v e. P. )
4		suplocsrlem.ss	. . . . . . . . 9 |- ( ph -> A C_ R. )
5		suplocsrlem.c	. . . . . . . . 9 |- ( ph -> C e. A )
6	4, 5	sseldd 3243	. . . . . . . 8 |- ( ph -> C e. R. )
7	6	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> C e. R. )
8		ltpsrprg 8134	. . . . . . 7 |- ( ( u e. P. /\ v e. P. /\ C e. R. ) -> ( ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) <-> u <P v ) )
9	2, 3, 7, 8	syl3anc 1274	. . . . . 6 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) <-> u <P v ) )
10	1, 9	mpbird 167	. . . . 5 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) )
11		breq2 4118	. . . . . . 7 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( ( C +R [ <. u , 1P >. ] ~R ) <R y <-> ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) ) )
12		breq2 4118	. . . . . . . . 9 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( z <R y <-> z <R ( C +R [ <. v , 1P >. ] ~R ) ) )
13	12	ralbidv 2544	. . . . . . . 8 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( A. z e. A z <R y <-> A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) )
14	13	orbi2d 798	. . . . . . 7 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R y ) <-> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) ) )
15	11, 14	imbi12d 234	. . . . . 6 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( ( ( C +R [ <. u , 1P >. ] ~R ) <R y -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R y ) ) <-> ( ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) ) ) )
16		breq1 4117	. . . . . . . . 9 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( x <R y <-> ( C +R [ <. u , 1P >. ] ~R ) <R y ) )
17		breq1 4117	. . . . . . . . . . 11 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( x <R z <-> ( C +R [ <. u , 1P >. ] ~R ) <R z ) )
18	17	rexbidv 2545	. . . . . . . . . 10 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( E. z e. A x <R z <-> E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z ) )
19	18	orbi1d 799	. . . . . . . . 9 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( ( E. z e. A x <R z \/ A. z e. A z <R y ) <-> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R y ) ) )
20	16, 19	imbi12d 234	. . . . . . . 8 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) <-> ( ( C +R [ <. u , 1P >. ] ~R ) <R y -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R y ) ) ) )
21	20	ralbidv 2544	. . . . . . 7 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) <-> A. y e. R. ( ( C +R [ <. u , 1P >. ] ~R ) <R y -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R y ) ) ) )
22		suplocsrlem.loc	. . . . . . . 8 |- ( ph -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )
23	22	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> A. x e. R. A. y e. R. ( x <R y -> ( E. z e. A x <R z \/ A. z e. A z <R y ) ) )
24		1pr 7885	. . . . . . . . . . . 12 |- 1P e. P.
25	24	a1i 9	. . . . . . . . . . 11 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> 1P e. P. )
26	2, 25	opelxpd 4787	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> <. u , 1P >. e. ( P. X. P. ) )
27		enrex 8068	. . . . . . . . . . 11 |- ~R e. _V
28	27	ecelqsi 6836	. . . . . . . . . 10 |- ( <. u , 1P >. e. ( P. X. P. ) -> [ <. u , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
29	26, 28	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> [ <. u , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
30		df-nr 8058	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
31	29, 30	eleqtrrdi 2328	. . . . . . . 8 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> [ <. u , 1P >. ] ~R e. R. )
32		addclsr 8084	. . . . . . . 8 |- ( ( C e. R. /\ [ <. u , 1P >. ] ~R e. R. ) -> ( C +R [ <. u , 1P >. ] ~R ) e. R. )
33	7, 31, 32	syl2anc 411	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( C +R [ <. u , 1P >. ] ~R ) e. R. )
34	21, 23, 33	rspcdva 2928	. . . . . 6 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> A. y e. R. ( ( C +R [ <. u , 1P >. ] ~R ) <R y -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R y ) ) )
35	3, 25	opelxpd 4787	. . . . . . . 8 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> <. v , 1P >. e. ( P. X. P. ) )
36	27	ecelqsi 6836	. . . . . . . . 9 |- ( <. v , 1P >. e. ( P. X. P. ) -> [ <. v , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
37	36, 30	eleqtrrdi 2328	. . . . . . . 8 |- ( <. v , 1P >. e. ( P. X. P. ) -> [ <. v , 1P >. ] ~R e. R. )
38	35, 37	syl 14	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> [ <. v , 1P >. ] ~R e. R. )
39		addclsr 8084	. . . . . . 7 |- ( ( C e. R. /\ [ <. v , 1P >. ] ~R e. R. ) -> ( C +R [ <. v , 1P >. ] ~R ) e. R. )
40	7, 38, 39	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( C +R [ <. v , 1P >. ] ~R ) e. R. )
41	15, 34, 40	rspcdva 2928	. . . . 5 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) ) )
42	10, 41	mpd 13	. . . 4 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) )
43	2	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> u e. P. )
44	7	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> C e. R. )
45		mappsrprg 8135	. . . . . . . . . . 11 |- ( ( u e. P. /\ C e. R. ) -> ( C +R -1R ) <R ( C +R [ <. u , 1P >. ] ~R ) )
46	43, 44, 45	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> ( C +R -1R ) <R ( C +R [ <. u , 1P >. ] ~R ) )
47		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> ( C +R [ <. u , 1P >. ] ~R ) <R z )
48		ltsosr 8095	. . . . . . . . . . 11 |- <R Or R.
49		ltrelsr 8069	. . . . . . . . . . 11 |- <R C_ ( R. X. R. )
50	48, 49	sotri 5163	. . . . . . . . . 10 |- ( ( ( C +R -1R ) <R ( C +R [ <. u , 1P >. ] ~R ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> ( C +R -1R ) <R z )
51	46, 47, 50	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> ( C +R -1R ) <R z )
52		map2psrprg 8136	. . . . . . . . . 10 |- ( C e. R. -> ( ( C +R -1R ) <R z <-> E. q e. P. ( C +R [ <. q , 1P >. ] ~R ) = z ) )
53	44, 52	syl 14	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> ( ( C +R -1R ) <R z <-> E. q e. P. ( C +R [ <. q , 1P >. ] ~R ) = z ) )
54	51, 53	mpbid 147	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> E. q e. P. ( C +R [ <. q , 1P >. ] ~R ) = z )
55		simpr 110	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> ( C +R [ <. q , 1P >. ] ~R ) = z )
56		simp-4r 544	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> z e. A )
57	55, 56	eqeltrd 2311	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> ( C +R [ <. q , 1P >. ] ~R ) e. A )
58		simpllr 536	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> ( C +R [ <. u , 1P >. ] ~R ) <R z )
59	58, 55	breqtrrd 4142	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. q , 1P >. ] ~R ) )
60	2	ad4antr 494	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> u e. P. )
61		simplr 529	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> q e. P. )
62	44	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> C e. R. )
63		ltpsrprg 8134	. . . . . . . . . . . . 13 |- ( ( u e. P. /\ q e. P. /\ C e. R. ) -> ( ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. q , 1P >. ] ~R ) <-> u <P q ) )
64	60, 61, 62, 63	syl3anc 1274	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> ( ( C +R [ <. u , 1P >. ] ~R ) <R ( C +R [ <. q , 1P >. ] ~R ) <-> u <P q ) )
65	59, 64	mpbid 147	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> u <P q )
66	57, 65	jca 306	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) /\ ( C +R [ <. q , 1P >. ] ~R ) = z ) -> ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) )
67	66	ex 115	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) /\ q e. P. ) -> ( ( C +R [ <. q , 1P >. ] ~R ) = z -> ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) ) )
68	67	reximdva 2646	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> ( E. q e. P. ( C +R [ <. q , 1P >. ] ~R ) = z -> E. q e. P. ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) ) )
69	54, 68	mpd 13	. . . . . . 7 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> E. q e. P. ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) )
70		opeq1 3888	. . . . . . . . . . . . . 14 |- ( w = q -> <. w , 1P >. = <. q , 1P >. )
71	70	eceq1d 6816	. . . . . . . . . . . . 13 |- ( w = q -> [ <. w , 1P >. ] ~R = [ <. q , 1P >. ] ~R )
72	71	oveq2d 6074	. . . . . . . . . . . 12 |- ( w = q -> ( C +R [ <. w , 1P >. ] ~R ) = ( C +R [ <. q , 1P >. ] ~R ) )
73	72	eleq1d 2303	. . . . . . . . . . 11 |- ( w = q -> ( ( C +R [ <. w , 1P >. ] ~R ) e. A <-> ( C +R [ <. q , 1P >. ] ~R ) e. A ) )
74		suplocsrlem.b	. . . . . . . . . . 11 |- B = { w e. P. | ( C +R [ <. w , 1P >. ] ~R ) e. A }
75	73, 74	elrab2 2979	. . . . . . . . . 10 |- ( q e. B <-> ( q e. P. /\ ( C +R [ <. q , 1P >. ] ~R ) e. A ) )
76	75	anbi1i 458	. . . . . . . . 9 |- ( ( q e. B /\ u <P q ) <-> ( ( q e. P. /\ ( C +R [ <. q , 1P >. ] ~R ) e. A ) /\ u <P q ) )
77		anass 401	. . . . . . . . 9 |- ( ( ( q e. P. /\ ( C +R [ <. q , 1P >. ] ~R ) e. A ) /\ u <P q ) <-> ( q e. P. /\ ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) ) )
78	76, 77	bitri 184	. . . . . . . 8 |- ( ( q e. B /\ u <P q ) <-> ( q e. P. /\ ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) ) )
79	78	rexbii2 2555	. . . . . . 7 |- ( E. q e. B u <P q <-> E. q e. P. ( ( C +R [ <. q , 1P >. ] ~R ) e. A /\ u <P q ) )
80	69, 79	sylibr 134	. . . . . 6 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ z e. A ) /\ ( C +R [ <. u , 1P >. ] ~R ) <R z ) -> E. q e. B u <P q )
81	80	rexlimdva2 2665	. . . . 5 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z -> E. q e. B u <P q ) )
82		breq1 4117	. . . . . . . . 9 |- ( z = ( C +R [ <. q , 1P >. ] ~R ) -> ( z <R ( C +R [ <. v , 1P >. ] ~R ) <-> ( C +R [ <. q , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) ) )
83		simplr 529	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) )
84		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> q e. B )
85	84, 75	sylib 122	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> ( q e. P. /\ ( C +R [ <. q , 1P >. ] ~R ) e. A ) )
86	85	simprd 114	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> ( C +R [ <. q , 1P >. ] ~R ) e. A )
87	82, 83, 86	rspcdva 2928	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> ( C +R [ <. q , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) )
88	85	simpld 112	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> q e. P. )
89	3	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> v e. P. )
90	7	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> C e. R. )
91		ltpsrprg 8134	. . . . . . . . 9 |- ( ( q e. P. /\ v e. P. /\ C e. R. ) -> ( ( C +R [ <. q , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) <-> q <P v ) )
92	88, 89, 90, 91	syl3anc 1274	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> ( ( C +R [ <. q , 1P >. ] ~R ) <R ( C +R [ <. v , 1P >. ] ~R ) <-> q <P v ) )
93	87, 92	mpbid 147	. . . . . . 7 |- ( ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) /\ q e. B ) -> q <P v )
94	93	ralrimiva 2617	. . . . . 6 |- ( ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) /\ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) -> A. q e. B q <P v )
95	94	ex 115	. . . . 5 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) -> A. q e. B q <P v ) )
96	81, 95	orim12d 794	. . . 4 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( ( E. z e. A ( C +R [ <. u , 1P >. ] ~R ) <R z \/ A. z e. A z <R ( C +R [ <. v , 1P >. ] ~R ) ) -> ( E. q e. B u <P q \/ A. q e. B q <P v ) ) )
97	42, 96	mpd 13	. . 3 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( E. q e. B u <P q \/ A. q e. B q <P v ) )
98	97	ex 115	. 2 |- ( ( ph /\ ( u e. P. /\ v e. P. ) ) -> ( u <P v -> ( E. q e. B u <P q \/ A. q e. B q <P v ) ) )
99	98	ralrimivva 2626	1 |- ( ph -> A. u e. P. A. v e. P. ( u <P v -> ( E. q e. B u <P q \/ A. q e. B q <P v ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   {crab 2526   C_ wss 3214   <.cop 3697   class class class wbr 4114   X. cxp 4752  (class class class)co 6058   [cec 6778   /.cqs 6779   P.cnp 7622   1Pc1p 7623   <P cltp 7626   ~R cer 7627   R.cnr 7628   -1Rcm1r 7631   +R cplr 7632   <R cltr 7634

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-i1p 7798  df-iplp 7799  df-imp 7800  df-iltp 7801  df-enr 8057  df-nr 8058  df-plr 8059  df-mr 8060  df-ltr 8061  df-0r 8062  df-1r 8063  df-m1r 8064

This theorem is referenced by:  suplocsrlempr  8138