Theorem suplocsrlemb 8163

Description: Lemma for suplocsr 8166. 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 541	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> u e. P. )
3		simplrr 542	. . . . . . 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 3249	. . . . . . . 8 |- ( ph -> C e. R. )
7	6	ad2antrr 492	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> C e. R. )
8		ltpsrprg 8160	. . . . . . 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 1278	. . . . . 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 4129	. . . . . . 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 4129	. . . . . . . . 9 |- ( y = ( C +R [ <. v , 1P >. ] ~R ) -> ( z <R y <-> z <R ( C +R [ <. v , 1P >. ] ~R ) ) )
13	12	ralbidv 2550	. . . . . . . 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 802	. . . . . . 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 4128	. . . . . . . . 9 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( x <R y <-> ( C +R [ <. u , 1P >. ] ~R ) <R y ) )
17		breq1 4128	. . . . . . . . . . 11 |- ( x = ( C +R [ <. u , 1P >. ] ~R ) -> ( x <R z <-> ( C +R [ <. u , 1P >. ] ~R ) <R z ) )
18	17	rexbidv 2551	. . . . . . . . . 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 803	. . . . . . . . 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 2550	. . . . . . 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 492	. . . . . . 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 7911	. . . . . . . . . . . 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 4802	. . . . . . . . . 10 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> <. u , 1P >. e. ( P. X. P. ) )
27		enrex 8094	. . . . . . . . . . 11 |- ~R e. _V
28	27	ecelqsi 6853	. . . . . . . . . 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 8084	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
31	29, 30	eleqtrrdi 2332	. . . . . . . 8 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> [ <. u , 1P >. ] ~R e. R. )
32		addclsr 8110	. . . . . . . 8 |- ( ( C e. R. /\ [ <. u , 1P >. ] ~R e. R. ) -> ( C +R [ <. u , 1P >. ] ~R ) e. R. )
33	7, 31, 32	syl2anc 415	. . . . . . 7 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( C +R [ <. u , 1P >. ] ~R ) e. R. )
34	21, 23, 33	rspcdva 2934	. . . . . 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 4802	. . . . . . . 8 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> <. v , 1P >. e. ( P. X. P. ) )
36	27	ecelqsi 6853	. . . . . . . . 9 |- ( <. v , 1P >. e. ( P. X. P. ) -> [ <. v , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
37	36, 30	eleqtrrdi 2332	. . . . . . . 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 8110	. . . . . . 7 |- ( ( C e. R. /\ [ <. v , 1P >. ] ~R e. R. ) -> ( C +R [ <. v , 1P >. ] ~R ) e. R. )
40	7, 38, 39	syl2anc 415	. . . . . 6 |- ( ( ( ph /\ ( u e. P. /\ v e. P. ) ) /\ u <P v ) -> ( C +R [ <. v , 1P >. ] ~R ) e. R. )
41	15, 34, 40	rspcdva 2934	. . . . 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 492	. . . . . . . . . . 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 492	. . . . . . . . . . 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 8161	. . . . . . . . . . 11 |- ( ( u e. P. /\ C e. R. ) -> ( C +R -1R ) <R ( C +R [ <. u , 1P >. ] ~R ) )
46	43, 44, 45	syl2anc 415	. . . . . . . . . 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 8121	. . . . . . . . . . 11 |- <R Or R.
49		ltrelsr 8095	. . . . . . . . . . 11 |- <R C_ ( R. X. R. )
50	48, 49	sotri 5178	. . . . . . . . . 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 415	. . . . . . . . 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 8162	. . . . . . . . . 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 548	. . . . . . . . . . . 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 2315	. . . . . . . . . . 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 540	. . . . . . . . . . . . 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 4153	. . . . . . . . . . . 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 498	. . . . . . . . . . . . 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 533	. . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 8160	. . . . . . . . . . . . 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 1278	. . . . . . . . . . . 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 2652	. . . . . . . 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 3899	. . . . . . . . . . . . . 14 |- ( w = q -> <. w , 1P >. = <. q , 1P >. )
71	70	eceq1d 6833	. . . . . . . . . . . . 13 |- ( w = q -> [ <. w , 1P >. ] ~R = [ <. q , 1P >. ] ~R )
72	71	oveq2d 6091	. . . . . . . . . . . 12 |- ( w = q -> ( C +R [ <. w , 1P >. ] ~R ) = ( C +R [ <. q , 1P >. ] ~R ) )
73	72	eleq1d 2307	. . . . . . . . . . 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 2985	. . . . . . . . . 10 |- ( q e. B <-> ( q e. P. /\ ( C +R [ <. q , 1P >. ] ~R ) e. A ) )
76	75	anbi1i 462	. . . . . . . . 9 |- ( ( q e. B /\ u <P q ) <-> ( ( q e. P. /\ ( C +R [ <. q , 1P >. ] ~R ) e. A ) /\ u <P q ) )
77		anass 405	. . . . . . . . 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 2561	. . . . . . 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 2671	. . . . 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 4128	. . . . . . . . 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 533	. . . . . . . . 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 2934	. . . . . . . 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 492	. . . . . . . . 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 492	. . . . . . . . 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 8160	. . . . . . . . 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 1278	. . . . . . . 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 2623	. . . . . 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 798	. . . 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 2632	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 720   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   <.cop 3708   class class class wbr 4125   X. cxp 4767  (class class class)co 6075   [cec 6795   /.cqs 6796   P.cnp 7648   1Pc1p 7649   <P cltp 7652   ~R cer 7653   R.cnr 7654   -1Rcm1r 7657   +R cplr 7658   <R cltr 7660

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-iplp 7825  df-imp 7826  df-iltp 7827  df-enr 8083  df-nr 8084  df-plr 8085  df-mr 8086  df-ltr 8087  df-0r 8088  df-1r 8089  df-m1r 8090

This theorem is referenced by:  suplocsrlempr  8164