Theorem suplocsrlemb 8016

Description: Lemma for suplocsr 8019. 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 535	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> u e. P. )
3		simplrr 536	. . . . . . 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 3226	. . . . . . . 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 8013	. . . . . . 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 1271	. . . . . 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 4090	. . . . . . 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 4090	. . . . . . . . 9 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( z <R y <-> z <R ( C +R [ <. v , 1P >. ] ~R ) ) )
13	12	ralbidv 2530	. . . . . . . 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 795	. . . . . . 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 4089	. . . . . . . . 9 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( x <R y <-> ( C +R [ <. u , 1P >. ] ~R ) <R y ) )
17		breq1 4089	. . . . . . . . . . 11 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( x <R z <-> ( C +R [ <. u , 1P >. ] ~R ) <R z ) )
18	17	rexbidv 2531	. . . . . . . . . 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 796	. . . . . . . . 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 2530	. . . . . . 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 7764	. . . . . . . . . . . 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 4756	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> <. u , 1P >. e. ( P. X. P. ) )
27		enrex 7947	. . . . . . . . . . 11 |- ~R e. _V
28	27	ecelqsi 6753	. . . . . . . . . 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 7937	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
31	29, 30	eleqtrrdi 2323	. . . . . . . 8 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> [ <. u , 1P >. ] ~R e. R. )
32		addclsr 7963	. . . . . . . 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 2913	. . . . . 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 4756	. . . . . . . 8 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> <. v , 1P >. e. ( P. X. P. ) )
36	27	ecelqsi 6753	. . . . . . . . 9 |- ( <. v , 1P >. e. ( P. X. P. ) -> [ <. v , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
37	36, 30	eleqtrrdi 2323	. . . . . . . 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 7963	. . . . . . 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 2913	. . . . 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 8014	. . . . . . . . . . 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 7974	. . . . . . . . . . 11 |- <R Or R.
49		ltrelsr 7948	. . . . . . . . . . 11 |- <R C_ ( R. X. R. )
50	48, 49	sotri 5130	. . . . . . . . . 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 8015	. . . . . . . . . 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 542	. . . . . . . . . . . 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 2306	. . . . . . . . . . 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 534	. . . . . . . . . . . . 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 4114	. . . . . . . . . . . 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 528	. . . . . . . . . . . . 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 8013	. . . . . . . . . . . . 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 1271	. . . . . . . . . . . 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 2632	. . . . . . . 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 3860	. . . . . . . . . . . . . 14 |- ( w = q -> <. w , 1P >. = <. q , 1P >. )
71	70	eceq1d 6733	. . . . . . . . . . . . 13 |- ( w = q -> [ <. w , 1P >. ] ~R = [ <. q , 1P >. ] ~R )
72	71	oveq2d 6029	. . . . . . . . . . . 12 |- ( w = q -> ( C +R [ <. w , 1P >. ] ~R ) = ( C +R [ <. q , 1P >. ] ~R ) )
73	72	eleq1d 2298	. . . . . . . . . . 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 2963	. . . . . . . . . 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 2541	. . . . . . 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 2651	. . . . 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 4089	. . . . . . . . 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 528	. . . . . . . . 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 2913	. . . . . . . 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 8013	. . . . . . . . 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 1271	. . . . . . . 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 2603	. . . . . 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 791	. . . 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 2612	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 713   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   {crab 2512   C_ wss 3198   <.cop 3670   class class class wbr 4086   X. cxp 4721  (class class class)co 6013   [cec 6695   /.cqs 6696   P.cnp 7501   1Pc1p 7502   <P cltp 7505   ~R cer 7506   R.cnr 7507   -1Rcm1r 7510   +R cplr 7511   <R cltr 7513

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563  df-enq0 7634  df-nq0 7635  df-0nq0 7636  df-plq0 7637  df-mq0 7638  df-inp 7676  df-i1p 7677  df-iplp 7678  df-imp 7679  df-iltp 7680  df-enr 7936  df-nr 7937  df-plr 7938  df-mr 7939  df-ltr 7940  df-0r 7941  df-1r 7942  df-m1r 7943

This theorem is referenced by:  suplocsrlempr  8017