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Theorem suplocexprlemloc 7717
Description: Lemma for suplocexpr 7721. 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 7411 . . . . 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 533 . . . . . . 7 |- ( ( ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) /\ q <Q r ) /\ ( v e. Q. /\ ( q <Q v /\ v <Q r ) ) ) -> ph )
5 simprl 529 . . . . . . . 8 |- ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) -> q e. Q. )
65ad2antrr 488 . . . . . . 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 529 . . . . . . 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 539 . . . . . 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 7590 . . . . . . . . 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 4006 . . . . . . . . . 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 4006 . . . . . . . . . . . 12 |- ( y = <. { l | l <Q v } , { u | v <Q u } >. -> ( z <P y <-> z <P <. { l | l <Q v } , { u | v <Q u } >. ) )
1413ralbidv 2477 . . . . . . . . . . 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 790 . . . . . . . . . 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 4005 . . . . . . . . . . . 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 4005 . . . . . . . . . . . . . 14 |- ( x = <. { l | l <Q q } , { u | q <Q u } >. -> ( x <P z <-> <. { l | l <Q q } , { u | q <Q u } >. <P z ) )
1918rexbidv 2478 . . . . . . . . . . . . 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 791 . . . . . . . . . . . 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 2477 . . . . . . . . . 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 488 . . . . . . . . . 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 535 . . . . . . . . . . 11 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> q e. Q. )
26 nqprlu 7543 . . . . . . . . . . 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 2846 . . . . . . . . 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 536 . . . . . . . . . 10 |- ( ( ( ph /\ ( q e. Q. /\ v e. Q. ) ) /\ q <Q v ) -> v e. Q. )
30 nqprlu 7543 . . . . . . . . . 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 2846 . . . . . . . 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 488 . . . . . . . . . . . . 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 7711 . . . . . . . . . . . . . . 15 |- ( ph -> A C_ P. )
3938ad4antr 494 . . . . . . . . . . . . . 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 528 . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . 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 7502 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . 14 |- w e. _V
46 breq2 4006 . . . . . . . . . . . . . 14 |- ( u = w -> ( q <Q u <-> q <Q w ) )
47 ltnqex 7545 . . . . . . . . . . . . . . 15 |- { l | l <Q q } e. _V
48 gtnqex 7546 . . . . . . . . . . . . . . 15 |- { u | q <Q u } e. _V
4947, 48op2nd 6145 . . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) = { u | q <Q u }
5045, 46, 49elab2 2885 . . . . . . . . . . . . 13 |- ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) <-> q <Q w )
5150anbi1i 458 . . . . . . . . . . . 12 |- ( ( w e. ( 2nd ` <. { l | l <Q q } , { u | q <Q u } >. ) /\ w e. ( 1st ` z ) ) <-> ( q <Q w /\ w e. ( 1st ` z ) ) )
5251rexbii 2484 . . . . . . . . . . 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 534 . . . . . . . . . . . . . 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 539 . . . . . . . . . . . . . . 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 7471 . . . . . . . . . . . . . . . . 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 540 . . . . . . . . . . . . . . . 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 7480 . . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` z ) , ( 2nd ` z ) >. e. P. /\ w e. ( 1st ` z ) ) -> ( q <Q w -> q e. ( 1st ` z ) ) )
6158, 59, 60syl2anc 411 . . . . . . . . . . . . . . 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 1591 . . . . . . . . . . . 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 2461 . . . . . . . . . . . 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 7709 . . . . . . . . . . 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 2599 . . . . . . . . 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 2597 . . . . . . . 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 6156 . . . . . . . . . . . . . . 15 |- 2nd : _V -onto-> _V
72 fofun 5438 . . . . . . . . . . . . . . 15 |- ( 2nd : _V -onto-> _V -> Fun 2nd )
7371, 72ax-mp 5 . . . . . . . . . . . . . 14 |- Fun 2nd
74 fvelima 5566 . . . . . . . . . . . . . 14 |- ( ( Fun 2nd /\ s e. ( 2nd " A ) ) -> E. t e. A ( 2nd ` t ) = s )
7573, 74mpan 424 . . . . . . . . . . . . 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 4005 . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . . . 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 7548 . . . . . . . . . . . . . . 15 |- ( ( v e. Q. /\ t e. P. ) -> ( v e. ( 2nd ` t ) <-> t <P <. { l | l <Q v } , { u | v <Q u } >. ) )
8581, 83, 84syl2anc 411 . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . 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 2256 . . . . . . . . . . . 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 2599 . . . . . . . . . . 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 2550 . . . . . . . . . 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 2740 . . . . . . . . . . 11 |- v e. _V
9291elint2 3851 . . . . . . . . . 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 786 . . . . . . 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 411 . . . . 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 4006 . . . . . . . . . 10 |- ( u = r -> ( w <Q u <-> w <Q r ) )
9998rexbidv 2478 . . . . . . . . 9 |- ( u = r -> ( E. w e. |^| ( 2nd " A ) w <Q u <-> E. w e. |^| ( 2nd " A ) w <Q r ) )
100 simprr 531 . . . . . . . . . 10 |- ( ( ph /\ ( q e. Q. /\ r e. Q. ) ) -> r e. Q. )
101100ad3antrrr 492 . . . . . . . . 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 540 . . . . . . . . . . 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 4005 . . . . . . . . . . 11 |- ( w = v -> ( w <Q r <-> v <Q r ) )
106105rspcev 2841 . . . . . . . . . 10 |- ( ( v e. |^| ( 2nd " A ) /\ v <Q r ) -> E. w e. |^| ( 2nd " A ) w <Q r )
107102, 104, 106syl2anc 411 . . . . . . . . 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 2895 . . . . . . . 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 7710 . . . . . . . . . . 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 2247 . . . . . . . . 9 |- ( ph -> ( r e. ( 2nd ` B ) <-> r e. { u e. Q. | E. w e. |^| ( 2nd " A ) w <Q u } ) )
113112ad4antr 494 . . . . . . . 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 788 . . . . 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 2599 . . 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 2559 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 708   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   _Vcvv 2737   C_ wss 3129   <.cop 3595   U.cuni 3809   |^|cint 3844   class class class wbr 4002   "cima 4628   Fun wfun 5209   -onto->wfo 5213   ` cfv 5215   1stc1st 6136   2ndc2nd 6137   Q.cnq 7276   <Q cltq 7281   P.cnp 7287   <P cltp 7291
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-eprel 4288  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-recs 6303  df-irdg 6368  df-1o 6414  df-oadd 6418  df-omul 6419  df-er 6532  df-ec 6534  df-qs 6538  df-ni 7300  df-pli 7301  df-mi 7302  df-lti 7303  df-plpq 7340  df-mpq 7341  df-enq 7343  df-nqqs 7344  df-plqqs 7345  df-mqqs 7346  df-1nqqs 7347  df-rq 7348  df-ltnqqs 7349  df-inp 7462  df-iltp 7466
This theorem is referenced by:  suplocexprlemex  7718
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