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Theorem aptiprlemu 7843
Description: Lemma for aptipr 7844. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprlemu |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
Proof of Theorem aptiprlemu
Dummy variables f g h s t u x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7678 . . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnminu 7692 . . . . . 6 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q x )
31, 2sylan 283 . . . . 5 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q x )
433ad2antl2 1184 . . . 4 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q x )
5 simprr 531 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> s <Q x )
6 ltexnqi 7612 . . . . . 6 |- ( s <Q x -> E. t e. Q. ( s +Q t ) = x )
75, 6syl 14 . . . . 5 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> E. t e. Q. ( s +Q t ) = x )
8 simpl1 1024 . . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> A e. P. )
98ad2antrr 488 . . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) -> A e. P. )
10 simprl 529 . . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) -> t e. Q. )
11 prop 7678 . . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12 prarloc2 7707 . . . . . . . 8 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1311, 12sylan 283 . . . . . . 7 |- ( ( A e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
149, 10, 13syl2anc 411 . . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
15 simpl2 1025 . . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> B e. P. )
1615ad3antrrr 492 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> B e. P. )
17 simpr 110 . . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> x e. ( 2nd ` B ) )
1817ad3antrrr 492 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> x e. ( 2nd ` B ) )
19 elprnqu 7685 . . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
201, 19sylan 283 . . . . . . . . 9 |- ( ( B e. P. /\ x e. ( 2nd ` B ) ) -> x e. Q. )
2116, 18, 20syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> x e. Q. )
228ad3antrrr 492 . . . . . . . . . 10 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> A e. P. )
23 simprl 529 . . . . . . . . . 10 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> u e. ( 1st ` A ) )
24 elprnql 7684 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
2511, 24sylan 283 . . . . . . . . . 10 |- ( ( A e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
2622, 23, 25syl2anc 411 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> u e. Q. )
2710adantr 276 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> t e. Q. )
28 addclnq 7578 . . . . . . . . 9 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
2926, 27, 28syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( u +Q t ) e. Q. )
30 nqtri3or 7599 . . . . . . . 8 |- ( ( x e. Q. /\ ( u +Q t ) e. Q. ) -> ( x <Q ( u +Q t ) \/ x = ( u +Q t ) \/ ( u +Q t ) <Q x ) )
3121, 29, 30syl2anc 411 . . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( x <Q ( u +Q t ) \/ x = ( u +Q t ) \/ ( u +Q t ) <Q x ) )
3215adantr 276 . . . . . . . . . . . . . 14 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> B e. P. )
33 simprl 529 . . . . . . . . . . . . . 14 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> s e. ( 2nd ` B ) )
34 elprnqu 7685 . . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ s e. ( 2nd ` B ) ) -> s e. Q. )
351, 34sylan 283 . . . . . . . . . . . . . 14 |- ( ( B e. P. /\ s e. ( 2nd ` B ) ) -> s e. Q. )
3632, 33, 35syl2anc 411 . . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> s e. Q. )
3736ad3antrrr 492 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> s e. Q. )
3833ad3antrrr 492 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> s e. ( 2nd ` B ) )
39 simplrr 536 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( s +Q t ) = x )
40 breq1 4086 . . . . . . . . . . . . . . . . 17 |- ( ( s +Q t ) = x -> ( ( s +Q t ) <Q ( u +Q t ) <-> x <Q ( u +Q t ) ) )
4140biimprd 158 . . . . . . . . . . . . . . . 16 |- ( ( s +Q t ) = x -> ( x <Q ( u +Q t ) -> ( s +Q t ) <Q ( u +Q t ) ) )
4239, 41syl 14 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( x <Q ( u +Q t ) -> ( s +Q t ) <Q ( u +Q t ) ) )
4342imp 124 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> ( s +Q t ) <Q ( u +Q t ) )
44 ltanqg 7603 . . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
4544adantl 277 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
4626adantr 276 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> u e. Q. )
4727adantr 276 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> t e. Q. )
48 addcomnqg 7584 . . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
4948adantl 277 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
5045, 37, 46, 47, 49caovord2d 6184 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> ( s <Q u <-> ( s +Q t ) <Q ( u +Q t ) ) )
5143, 50mpbird 167 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> s <Q u )
5222adantr 276 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> A e. P. )
5323adantr 276 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> u e. ( 1st ` A ) )
54 prcdnql 7687 . . . . . . . . . . . . . . 15 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> ( s <Q u -> s e. ( 1st ` A ) ) )
5511, 54sylan 283 . . . . . . . . . . . . . 14 |- ( ( A e. P. /\ u e. ( 1st ` A ) ) -> ( s <Q u -> s e. ( 1st ` A ) ) )
5652, 53, 55syl2anc 411 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> ( s <Q u -> s e. ( 1st ` A ) ) )
5751, 56mpd 13 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> s e. ( 1st ` A ) )
58 rspe 2579 . . . . . . . . . . . 12 |- ( ( s e. Q. /\ ( s e. ( 2nd ` B ) /\ s e. ( 1st ` A ) ) ) -> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` A ) ) )
5937, 38, 57, 58syl12anc 1269 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` A ) ) )
6016adantr 276 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> B e. P. )
61 ltdfpr 7709 . . . . . . . . . . . 12 |- ( ( B e. P. /\ A e. P. ) -> ( B <P A <-> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` A ) ) ) )
6260, 52, 61syl2anc 411 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> ( B <P A <-> E. s e. Q. ( s e. ( 2nd ` B ) /\ s e. ( 1st ` A ) ) ) )
6359, 62mpbird 167 . . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> B <P A )
64 simpll3 1062 . . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> -. B <P A )
6564ad3antrrr 492 . . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> -. B <P A )
6663, 65pm2.21dd 623 . . . . . . . . 9 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ x <Q ( u +Q t ) ) -> x e. ( 2nd ` A ) )
6766ex 115 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( x <Q ( u +Q t ) -> x e. ( 2nd ` A ) ) )
68 simprr 531 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( u +Q t ) e. ( 2nd ` A ) )
69 eleq1 2292 . . . . . . . . 9 |- ( x = ( u +Q t ) -> ( x e. ( 2nd ` A ) <-> ( u +Q t ) e. ( 2nd ` A ) ) )
7068, 69syl5ibrcom 157 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( x = ( u +Q t ) -> x e. ( 2nd ` A ) ) )
71 prcunqu 7688 . . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( u +Q t ) e. ( 2nd ` A ) ) -> ( ( u +Q t ) <Q x -> x e. ( 2nd ` A ) ) )
7211, 71sylan 283 . . . . . . . . 9 |- ( ( A e. P. /\ ( u +Q t ) e. ( 2nd ` A ) ) -> ( ( u +Q t ) <Q x -> x e. ( 2nd ` A ) ) )
7322, 68, 72syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) <Q x -> x e. ( 2nd ` A ) ) )
7467, 70, 733jaod 1338 . . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( x <Q ( u +Q t ) \/ x = ( u +Q t ) \/ ( u +Q t ) <Q x ) -> x e. ( 2nd ` A ) ) )
7531, 74mpd 13 . . . . . 6 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> x e. ( 2nd ` A ) )
7614, 75rexlimddv 2653 . . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) /\ ( t e. Q. /\ ( s +Q t ) = x ) ) -> x e. ( 2nd ` A ) )
777, 76rexlimddv 2653 . . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) /\ ( s e. ( 2nd ` B ) /\ s <Q x ) ) -> x e. ( 2nd ` A ) )
784, 77rexlimddv 2653 . . 3 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> x e. ( 2nd ` A ) )
7978ex 115 . 2 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( x e. ( 2nd ` B ) -> x e. ( 2nd ` A ) ) )
8079ssrdv 3230 1 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 2nd ` B ) C_ ( 2nd ` A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1001   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   <.cop 3669   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   1stc1st 6293   2ndc2nd 6294   Q.cnq 7483   +Q cplq 7485   <Q cltq 7488   P.cnp 7494   <P cltp 7498
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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681
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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4381  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-1o 6573  df-2o 6574  df-oadd 6577  df-omul 6578  df-er 6693  df-ec 6695  df-qs 6699  df-ni 7507  df-pli 7508  df-mi 7509  df-lti 7510  df-plpq 7547  df-mpq 7548  df-enq 7550  df-nqqs 7551  df-plqqs 7552  df-mqqs 7553  df-1nqqs 7554  df-rq 7555  df-ltnqqs 7556  df-enq0 7627  df-nq0 7628  df-0nq0 7629  df-plq0 7630  df-mq0 7631  df-inp 7669  df-iltp 7673
This theorem is referenced by:  aptipr  7844  suplocexprlemmu  7921
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