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Theorem aptiprlemu 7702
Description: Lemma for aptipr 7703. (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 7537 . . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnminu 7551 . . . . . 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 1162 . . . 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 7471 . . . . . 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 1002 . . . . . . . 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 7537 . . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12 prarloc2 7566 . . . . . . . 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 1003 . . . . . . . . . 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 7544 . . . . . . . . . 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 7543 . . . . . . . . . . 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 7437 . . . . . . . . 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 7458 . . . . . . . 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 7544 . . . . . . . . . . . . . . 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 4033 . . . . . . . . . . . . . . . . 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 7462 . . . . . . . . . . . . . . . 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 7443 . . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . 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 7546 . . . . . . . . . . . . . . 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 2543 . . . . . . . . . . . 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 1247 . . . . . . . . . . 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 7568 . . . . . . . . . . . 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 1040 . . . . . . . . . . 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 621 . . . . . . . . 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 2256 . . . . . . . . 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 7547 . . . . . . . . . 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 1315 . . . . . . 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 2616 . . . . 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 2616 . . . 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 2616 . . 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 3186 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 979   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   C_ wss 3154   <.cop 3622   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   +Q cplq 7344   <Q cltq 7347   P.cnp 7353   <P cltp 7357
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iltp 7532
This theorem is referenced by:  aptipr  7703  suplocexprlemmu  7780
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