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Theorem aptiprlemu 7788
Description: Lemma for aptipr 7789. (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 7623 . . . . . 6 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnminu 7637 . . . . . 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 1163 . . . 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 7557 . . . . . 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 1003 . . . . . . . 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 7623 . . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12 prarloc2 7652 . . . . . . . 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 1004 . . . . . . . . . 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 7630 . . . . . . . . . 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 7629 . . . . . . . . . . 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 7523 . . . . . . . . 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 7544 . . . . . . . 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 7630 . . . . . . . . . . . . . . 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 4062 . . . . . . . . . . . . . . . . 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 7548 . . . . . . . . . . . . . . . 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 7529 . . . . . . . . . . . . . . . 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 6139 . . . . . . . . . . . . . 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 7632 . . . . . . . . . . . . . . 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 2557 . . . . . . . . . . . 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 1248 . . . . . . . . . . 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 7654 . . . . . . . . . . . 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 1041 . . . . . . . . . . 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 2270 . . . . . . . . 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 7633 . . . . . . . . . 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 1317 . . . . . . 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 2630 . . . . 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 2630 . . . 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 2630 . . 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 3207 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 980   /\ w3a 981   = wceq 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   <.cop 3646   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   +Q cplq 7430   <Q cltq 7433   P.cnp 7439   <P cltp 7443
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654
This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-iltp 7618
This theorem is referenced by:  aptipr  7789  suplocexprlemmu  7866
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