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Theorem aptiprlemu 7997
Description: Lemma for aptipr 7998. (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 7832 . . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnminu 7846 . . . . . 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 1191 . . . 4 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> E. s e. ( 2nd ` B ) s <Q x )
5 simprr 537 . . . . . 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 7766 . . . . . 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 1031 . . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> A e. P. )
98ad2antrr 492 . . . . . . 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 535 . . . . . . 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 7832 . . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12 prarloc2 7861 . . . . . . . 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 415 . . . . . 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 1032 . . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. /\ -. B <P A ) /\ x e. ( 2nd ` B ) ) -> B e. P. )
1615ad3antrrr 496 . . . . . . . . 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 496 . . . . . . . . 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 7839 . . . . . . . . . 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 415 . . . . . . . 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 496 . . . . . . . . . 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 535 . . . . . . . . . 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 7838 . . . . . . . . . . 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 415 . . . . . . . . 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 7732 . . . . . . . . 9 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
2926, 27, 28syl2anc 415 . . . . . . . 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 7753 . . . . . . . 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 415 . . . . . . 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 535 . . . . . . . . . . . . . 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 7839 . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . 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 496 . . . . . . . . . . . 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 496 . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . 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 4128 . . . . . . . . . . . . . . . . 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 7757 . . . . . . . . . . . . . . . 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 7738 . . . . . . . . . . . . . . . 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 6249 . . . . . . . . . . . . . 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 7841 . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . 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 2599 . . . . . . . . . . . 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 1276 . . . . . . . . . . 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 7863 . . . . . . . . . . . 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 415 . . . . . . . . . . 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 1069 . . . . . . . . . . 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 496 . . . . . . . . . 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 629 . . . . . . . . 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 537 . . . . . . . . 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 2301 . . . . . . . . 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 7842 . . . . . . . . . 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 415 . . . . . . . 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 1345 . . . . . . 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 2673 . . . . 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 2673 . . . 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 2673 . . 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 3254 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 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   +Q cplq 7639   <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-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-iltp 7827
This theorem is referenced by:  aptipr  7998  suplocexprlemmu  8075
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