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Theorem suplocexprlemloc 8078
Description: Lemma for suplocexpr 8082. The putative supremum is located. (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 ) ) )
suplocexpr.b |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
Assertion
Ref Expression
suplocexprlemloc |- ( ph -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) )
Distinct variable groups:   u, A, z, w   x, A, y, u, z   u, q, z, w   x, q, y, ph   ph, r, w, q   ph, z, x, y   u, r
Allowed substitution hints:   ph( u)   A( r, q)   B( x, y, z, w, u, r, q)
Proof of Theorem suplocexprlemloc
Dummy variables s t v l are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 110 . . . . 5 |- ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) -> q <Q r )
2 ltbtwnnqq 7772 . . . . 5 |- ( q <Q r <-> E. v e. Q. ( q <Q v /\ v <Q r ) )
31, 2sylib 122 . . . 4 |- ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) -> E. v e. Q. ( q <Q v /\ v <Q r ) )
4 simplll 539 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ph )
5 simprl 535 . . . . . . . 8 |- ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) -> q e. Q. )
65ad2antrr 492 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> q e. Q. )
7 simprl 535 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> v e. Q. )
84, 6, 7jca32 310 . . . . . 6 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ( ph /\ ( q e. Q. /\ v e. Q. ) ) )
9 simprrl 545 . . . . . 6 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> q <Q v )
10 ltnqpri 7951 . . . . . . . . 9 |- ( q <Q v -> <. { l | l <Q q } , { u | q <Q u } >. <P <. { l | l <Q v } , { u | v <Q u } >. )
1110adantl 277 . . . . . . . 8 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> <. { l | l <Q q } , { u | q <Q u } >. <P <. { l | l <Q v } , { u | v <Q u } >. )
12 breq2 4129 . . . . . . . . . 10 |- ( y = <. { l | l <Q v } , { u | v <Q u } >. -> ( <. { l | l <Q q } , { u | q <Q u } >. <P y <-> <. { l | l <Q q } , { u | q <Q u } >. <P <. { l | l <Q v } , { u | v <Q u } >. ) )
13 breq2 4129 . . . . . . . . . . . 12 |- ( y = <. { l | l <Q v } , { u | v <Q u } >. -> ( z <P y <-> z <P <. { l | l <Q v } , { u | v <Q u } >. ) )
1413ralbidv 2550 . . . . . . . . . . 11 |- ( y = <. { l | l <Q v } , { u | v <Q u } >. -> ( A. z e. A z <P y <-> A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) )
1514orbi2d 802 . . . . . . . . . 10 |- ( y = <. { l | l <Q v } , { u | v <Q u } >. -> ( ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P y ) <-> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) ) )
1612, 15imbi12d 234 . . . . . . . . 9 |- ( y = <. { l | l <Q v } , { u | v <Q u } >. -> ( ( <. { l | l <Q q } , { u | q <Q u } >. <P y -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P y ) ) <-> ( <. { l | l <Q q } , { u | q <Q u } >. <P <. { l | l <Q v } , { u | v <Q u } >. -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) ) ) )
17 breq1 4128 . . . . . . . . . . . 12 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( x <P y <-> <. { l | l <Q q } , { u | q <Q u } >. <P y ) )
18 breq1 4128 . . . . . . . . . . . . . 14 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( x <P z <-> <. { l | l <Q q } , { u | q <Q u } >. <P z ) )
1918rexbidv 2551 . . . . . . . . . . . . 13 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( E. z e. A x <P z <-> E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z ) )
2019orbi1d 803 . . . . . . . . . . . 12 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( ( E. z e. A x <P z \/ A. z e. A z <P y ) <-> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P y ) ) )
2117, 20imbi12d 234 . . . . . . . . . . 11 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) <-> ( <. { l | l <Q q } , { u | q <Q u } >. <P y -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P y ) ) ) )
2221ralbidv 2550 . . . . . . . . . 10 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( A. y e. P. ( x <P y -> ( E. z e. A x <P z \/ A. z e. A z <P y ) ) <-> A. y e. P. ( <. { l | l <Q q } , { u | q <Q u } >. <P y -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P y ) ) ) )
23 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 ) ) )
2423ad2antrr 492 . . . . . . . . . 10 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> 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 ) ) )
25 simplrl 541 . . . . . . . . . . 11 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> q e. Q. )
26 nqprlu 7904 . . . . . . . . . . 11 |- ( q e. Q. -> <. { l | l <Q q } , { u | q <Q u } >. e. P. )
2725, 26syl 14 . . . . . . . . . 10 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> <. { l | l <Q q } , { u | q <Q u } >. e. P. )
2822, 24, 27rspcdva 2934 . . . . . . . . 9 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> A. y e. P. ( <. { l | l <Q q } , { u | q <Q u } >. <P y -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P y ) ) )
29 simplrr 542 . . . . . . . . . 10 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> v e. Q. )
30 nqprlu 7904 . . . . . . . . . 10 |- ( v e. Q. -> <. { l | l <Q v } , { u | v <Q u } >. e. P. )
3129, 30syl 14 . . . . . . . . 9 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> <. { l | l <Q v } , { u | v <Q u } >. e. P. )
3216, 28, 31rspcdva 2934 . . . . . . . 8 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> ( <. { l | l <Q q } , { u | q <Q u } >. <P <. { l | l <Q v } , { u | v <Q u } >. -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) ) )
3311, 32mpd 13 . . . . . . 7 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) )
34 simpr 110 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> <. { l | l <Q q } , { u | q <Q u } >. <P z )
3527ad2antrr 492 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> <. { l | l <Q q } , { u | q <Q u } >. e. P. )
36 suplocexpr.m . . . . . . . . . . . . . . . 16 |- ( ph -> E. x x e. A )
37 suplocexpr.ub . . . . . . . . . . . . . . . 16 |- ( ph -> E. x e. P. A. y e. A y <P x )
3836, 37, 23suplocexprlemss 8072 . . . . . . . . . . . . . . 15 |- ( ph -> A C_ P. )
3938ad4antr 498 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> A C_ P. )
40 simplr 533 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> z e. A )
4139, 40sseldd 3249 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> z e. P. )
42 ltdfpr 7863 . . . . . . . . . . . . 13 |- ( ( <. { l | l <Q q } , { u | q <Q u } >. e. P. /\ z e. P. ) -> ( <. { l | l <Q q } , { u | q <Q u } >. <P z <-> E. w e. Q. ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) /\ w e. ( 1st ` z ) ) ) )
4335, 41, 42syl2anc 415 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> ( <. { l | l <Q q } , { u | q <Q u } >. <P z <-> E. w e. Q. ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) /\ w e. ( 1st ` z ) ) ) )
4434, 43mpbid 147 . . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> E. w e. Q. ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) /\ w e. ( 1st ` z ) ) )
45 vex 2824 . . . . . . . . . . . . . 14 |- w e. _V
46 breq2 4129 . . . . . . . . . . . . . 14 |- ( u = w -> ( q <Q u <-> q <Q w ) )
47 ltnqex 7906 . . . . . . . . . . . . . . 15 |- { l | l <Q q } e. _V
48 gtnqex 7907 . . . . . . . . . . . . . . 15 |- { u | q <Q u } e. _V
4947, 48op2nd 6371 . . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) = { u | q <Q u }
5045, 46, 49elab2 2974 . . . . . . . . . . . . 13 |- ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) <-> q <Q w )
5150anbi1i 462 . . . . . . . . . . . 12 |- ( ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) /\ w e. ( 1st ` z ) ) <-> ( q <Q w /\ w e. ( 1st ` z ) ) )
5251rexbii 2557 . . . . . . . . . . 11 |- ( E. w e. Q. ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) /\ w e. ( 1st ` z ) ) <-> E. w e. Q. ( q <Q w /\ w e. ( 1st ` z ) ) )
5344, 52sylib 122 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> E. w e. Q. ( q <Q w /\ w e. ( 1st ` z ) ) )
54 simpllr 540 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> z e. A )
55 simprrl 545 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> q <Q w )
5641adantr 276 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> z e. P. )
57 prop 7832 . . . . . . . . . . . . . . . . 17 |- ( z e. P. -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. )
5856, 57syl 14 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. )
59 simprrr 546 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> w e. ( 1st ` z ) )
60 prcdnql 7841 . . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. /\ w e. ( 1st ` z ) ) -> ( q <Q w -> q e. ( 1st ` z ) ) )
6158, 59, 60syl2anc 415 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> ( q <Q w -> q e. ( 1st ` z ) ) )
6255, 61mpd 13 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> q e. ( 1st ` z ) )
6354, 62jca 306 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> ( z e. A /\ q e. ( 1st ` z ) ) )
646319.8ad 1644 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> E. z ( z e. A /\ q e. ( 1st ` z ) ) )
65 df-rex 2534 . . . . . . . . . . . 12 |- ( E. z e. A q e. ( 1st ` z ) <-> E. z ( z e. A /\ q e. ( 1st ` z ) ) )
6664, 65sylibr 134 . . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> E. z e. A q e. ( 1st ` z ) )
67 suplocexprlemell 8070 . . . . . . . . . . 11 |- ( q e. U. ( 1st " A ) <-> E. z e. A q e. ( 1st ` z ) )
6866, 67sylibr 134 . . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) /\ ( w e. Q. /\ ( q <Q w /\ w e. ( 1st ` z ) ) ) ) -> q e. U. ( 1st " A ) )
6953, 68rexlimddv 2673 . . . . . . . . 9 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ z e. A ) /\ <. { l | l <Q q } , { u | q <Q u } >. <P z ) -> q e. U. ( 1st " A ) )
7069rexlimdva2 2671 . . . . . . . 8 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z -> q e. U. ( 1st " A ) ) )
71 fo2nd 6382 . . . . . . . . . . . . . . 15 |- 2nd : _V -onto-> _V
72 fofun 5611 . . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
7371, 72ax-mp 5 . . . . . . . . . . . . . 14 |- Fun 2nd
74 fvelima 5748 . . . . . . . . . . . . . 14 |- ( ( Fun 2nd /\ s e. ( 2nd " A ) ) -> E. t e. A ( 2nd ` t ) = s )
7573, 74mpan 428 . . . . . . . . . . . . 13 |- ( s e. ( 2nd " A ) -> E. t e. A ( 2nd ` t ) = s )
7675adantl 277 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) -> E. t e. A ( 2nd ` t ) = s )
77 breq1 4128 . . . . . . . . . . . . . . 15 |- ( z = t -> ( z <P <. { l | l <Q v } , { u | v <Q u } >. <-> t <P <. { l | l <Q v } , { u | v <Q u } >. ) )
78 simpllr 540 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. )
79 simprl 535 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> t e. A )
8077, 78, 79rspcdva 2934 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> t <P <. { l | l <Q v } , { u | v <Q u } >. )
8129ad3antrrr 496 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> v e. Q. )
8238ad5antr 500 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> A C_ P. )
8382, 79sseldd 3249 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> t e. P. )
84 nqpru 7909 . . . . . . . . . . . . . . 15 |- ( ( v e. Q. /\ t e. P. ) -> ( v e. ( 2nd ` t ) <-> t <P <. { l | l <Q v } , { u | v <Q u } >. ) )
8581, 83, 84syl2anc 415 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> ( v e. ( 2nd ` t ) <-> t <P <. { l | l <Q v } , { u | v <Q u } >. ) )
8680, 85mpbird 167 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> v e. ( 2nd ` t ) )
87 simprr 537 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> ( 2nd ` t ) = s )
8886, 87eleqtrd 2317 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) /\ ( t e. A /\ ( 2nd ` t ) = s ) ) -> v e. s )
8976, 88rexlimddv 2673 . . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) /\ s e. ( 2nd " A ) ) -> v e. s )
9089ralrimiva 2623 . . . . . . . . . 10 |- ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) -> A. s e. ( 2nd " A ) v e. s )
91 vex 2824 . . . . . . . . . . 11 |- v e. _V
9291elint2 3972 . . . . . . . . . 10 |- ( v e. |^| ( 2nd " A ) <-> A. s e. ( 2nd " A ) v e. s )
9390, 92sylibr 134 . . . . . . . . 9 |- ( ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) /\ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) -> v e. |^| ( 2nd " A ) )
9493ex 115 . . . . . . . 8 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> ( A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. -> v e. |^| ( 2nd " A ) ) )
9570, 94orim12d 798 . . . . . . 7 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> ( ( E. z e. A <. { l | l <Q q } , { u | q <Q u } >. <P z \/ A. z e. A z <P <. { l | l <Q v } , { u | v <Q u } >. ) -> ( q e. U. ( 1st " A ) \/ v e. |^| ( 2nd " A ) ) ) )
9633, 95mpd 13 . . . . . 6 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> ( q e. U. ( 1st " A ) \/ v e. |^| ( 2nd " A ) ) )
978, 9, 96syl2anc 415 . . . . 5 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ( q e. U. ( 1st " A ) \/ v e. |^| ( 2nd " A ) ) )
98 breq2 4129 . . . . . . . . . 10 |- ( u = r -> ( w <Q u <-> w <Q r ) )
9998rexbidv 2551 . . . . . . . . 9 |- ( u = r -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q r ) )
100 simprr 537 . . . . . . . . . 10 |- ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) -> r e. Q. )
101100ad3antrrr 496 . . . . . . . . 9 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> r e. Q. )
102 simpr 110 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> v e. |^| ( 2nd " A ) )
103 simprrr 546 . . . . . . . . . . 11 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> v <Q r )
104103adantr 276 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> v <Q r )
105 breq1 4128 . . . . . . . . . . 11 |- ( w = v -> ( w <Q r <-> v <Q r ) )
106105rspcev 2929 . . . . . . . . . 10 |- ( ( v e. |^| ( 2nd " A ) /\ v <Q r ) -> E. w e. |^| ( 2nd " A ) w <Q r )
107102, 104, 106syl2anc 415 . . . . . . . . 9 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> E. w e. |^| ( 2nd " A ) w <Q r )
10899, 101, 107elrabd 2984 . . . . . . . 8 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
109 suplocexpr.b . . . . . . . . . . . 12 |- B = <. U. ( 1st " A ) , { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } >.
110109suplocexprlem2b 8071 . . . . . . . . . . 11 |- ( A C_ P. -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
11138, 110syl 14 . . . . . . . . . 10 |- ( ph -> ( 2nd ` B ) = { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } )
112111eleq2d 2308 . . . . . . . . 9 |- ( ph -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
113112ad4antr 498 . . . . . . . 8 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
114108, 113mpbird 167 . . . . . . 7 |- ( ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) /\ v e. |^| ( 2nd " A ) ) -> r e. ( 2nd ` B ) )
115114ex 115 . . . . . 6 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ( v e. |^| ( 2nd " A ) -> r e. ( 2nd ` B ) ) )
116115orim2d 800 . . . . 5 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ( ( q e. U. ( 1st " A ) \/ v e. |^| ( 2nd " A ) ) -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) )
11797, 116mpd 13 . . . 4 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) )
1183, 117rexlimddv 2673 . . 3 |- ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) )
119118ex 115 . 2 |- ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) -> ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) )
120119ralrimivva 2632 1 |- ( ph -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. U. ( 1st " A ) \/ r e. ( 2nd ` B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   <.cop 3708   U.cuni 3930   |^|cint 3965   class class class wbr 4125   "cima 4772   Fun wfun 5366   -onto->wfo 5370   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   <Q cltq 7642   P.cnp 7648   <P cltp 7652
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-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-inp 7823  df-iltp 7827
This theorem is referenced by:  suplocexprlemex  8079
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