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Theorem aptiprleml 8006
Description: Lemma for aptipr 8008. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprleml |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
Proof of Theorem aptiprleml
Dummy variables f g h s t u v x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7842 . . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 prnmaxl 7855 . . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> E. s e. ( 1st ` A ) x <Q s )
31, 2sylan 283 . . . . . 6 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> E. s e. ( 1st ` A ) x <Q s )
43ad2ant2rl 515 . . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> E. s e. ( 1st ` A ) x <Q s )
5 ltexnqi 7776 . . . . . . 7 |- ( x <Q s -> E. t e. Q. ( x +Q t ) = s )
65ad2antll 495 . . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) -> E. t e. Q. ( x +Q t ) = s )
7 simplr 533 . . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> B e. P. )
87ad2antrr 492 . . . . . . . 8 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) -> B e. P. )
9 simprl 535 . . . . . . . 8 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) -> t e. Q. )
10 prop 7842 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
11 prarloc2 7871 . . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` B ) ( u +Q t ) e. ( 2nd ` B ) )
1210, 11sylan 283 . . . . . . . 8 |- ( ( B e. P. /\ t e. Q. ) -> E. u e. ( 1st ` B ) ( u +Q t ) e. ( 2nd ` B ) )
138, 9, 12syl2anc 415 . . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) -> E. u e. ( 1st ` B ) ( u +Q t ) e. ( 2nd ` B ) )
148adantr 276 . . . . . . . . . 10 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> B e. P. )
15 simprl 535 . . . . . . . . . 10 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> u e. ( 1st ` B ) )
16 elprnql 7848 . . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ u e. ( 1st ` B ) ) -> u e. Q. )
1710, 16sylan 283 . . . . . . . . . 10 |- ( ( B e. P. /\ u e. ( 1st ` B ) ) -> u e. Q. )
1814, 15, 17syl2anc 415 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> u e. Q. )
19 simpll 531 . . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> A e. P. )
2019ad3antrrr 496 . . . . . . . . . 10 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> A e. P. )
21 simprr 537 . . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> x e. ( 1st ` A ) )
2221ad3antrrr 496 . . . . . . . . . 10 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> x e. ( 1st ` A ) )
23 elprnql 7848 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
241, 23sylan 283 . . . . . . . . . 10 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
2520, 22, 24syl2anc 415 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> x e. Q. )
26 nqtri3or 7763 . . . . . . . . 9 |- ( ( u e. Q. /\ x e. Q. ) -> ( u <Q x \/ u = x \/ x <Q u ) )
2718, 25, 26syl2anc 415 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u <Q x \/ u = x \/ x <Q u ) )
2818adantr 276 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> u e. Q. )
29 simplrl 541 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> t e. Q. )
3029adantr 276 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> t e. Q. )
31 addclnq 7742 . . . . . . . . . . . . . 14 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
3228, 30, 31syl2anc 415 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> ( u +Q t ) e. Q. )
33 ltanqg 7767 . . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
3433adantl 277 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
35 addcomnqg 7748 . . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
3635adantl 277 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
3734, 18, 25, 29, 36caovord2d 6259 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u <Q x <-> ( u +Q t ) <Q ( x +Q t ) ) )
38 simplrr 542 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( x +Q t ) = s )
39 simprl 535 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) -> s e. ( 1st ` A ) )
4039ad2antrr 492 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> s e. ( 1st ` A ) )
4138, 40eqeltrd 2315 . . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( x +Q t ) e. ( 1st ` A ) )
42 prcdnql 7851 . . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( x +Q t ) e. ( 1st ` A ) ) -> ( ( u +Q t ) <Q ( x +Q t ) -> ( u +Q t ) e. ( 1st ` A ) ) )
431, 42sylan 283 . . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ ( x +Q t ) e. ( 1st ` A ) ) -> ( ( u +Q t ) <Q ( x +Q t ) -> ( u +Q t ) e. ( 1st ` A ) ) )
4420, 41, 43syl2anc 415 . . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( ( u +Q t ) <Q ( x +Q t ) -> ( u +Q t ) e. ( 1st ` A ) ) )
4537, 44sylbid 150 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u <Q x -> ( u +Q t ) e. ( 1st ` A ) ) )
46 simprr 537 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u +Q t ) e. ( 2nd ` B ) )
4745, 46jctild 316 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u <Q x -> ( ( u +Q t ) e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 1st ` A ) ) ) )
4847imp 124 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> ( ( u +Q t ) e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 1st ` A ) ) )
49 eleq1 2301 . . . . . . . . . . . . . . 15 |- ( v = ( u +Q t ) -> ( v e. ( 2nd ` B ) <-> ( u +Q t ) e. ( 2nd ` B ) ) )
50 eleq1 2301 . . . . . . . . . . . . . . 15 |- ( v = ( u +Q t ) -> ( v e. ( 1st ` A ) <-> ( u +Q t ) e. ( 1st ` A ) ) )
5149, 50anbi12d 477 . . . . . . . . . . . . . 14 |- ( v = ( u +Q t ) -> ( ( v e. ( 2nd ` B ) /\ v e. ( 1st ` A ) ) <-> ( ( u +Q t ) e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 1st ` A ) ) ) )
5251rspcev 2929 . . . . . . . . . . . . 13 |- ( ( ( u +Q t ) e. Q. /\ ( ( u +Q t ) e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 1st ` A ) ) ) -> E. v e. Q. ( v e. ( 2nd ` B ) /\ v e. ( 1st ` A ) ) )
5332, 48, 52syl2anc 415 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> E. v e. Q. ( v e. ( 2nd ` B ) /\ v e. ( 1st ` A ) ) )
54 ltdfpr 7873 . . . . . . . . . . . . . 14 |- ( ( B e. P. /\ A e. P. ) -> ( B <P A <-> E. v e. Q. ( v e. ( 2nd ` B ) /\ v e. ( 1st ` A ) ) ) )
5514, 20, 54syl2anc 415 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( B <P A <-> E. v e. Q. ( v e. ( 2nd ` B ) /\ v e. ( 1st ` A ) ) ) )
5655adantr 276 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> ( B <P A <-> E. v e. Q. ( v e. ( 2nd ` B ) /\ v e. ( 1st ` A ) ) ) )
5753, 56mpbird 167 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> B <P A )
58 simplrl 541 . . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) -> -. B <P A )
5958ad3antrrr 496 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> -. B <P A )
6057, 59pm2.21dd 629 . . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) /\ u <Q x ) -> x e. ( 1st ` B ) )
6160ex 115 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u <Q x -> x e. ( 1st ` B ) ) )
62 eleq1 2301 . . . . . . . . . 10 |- ( u = x -> ( u e. ( 1st ` B ) <-> x e. ( 1st ` B ) ) )
6315, 62syl5ibcom 155 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( u = x -> x e. ( 1st ` B ) ) )
64 prcdnql 7851 . . . . . . . . . . 11 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ u e. ( 1st ` B ) ) -> ( x <Q u -> x e. ( 1st ` B ) ) )
6510, 64sylan 283 . . . . . . . . . 10 |- ( ( B e. P. /\ u e. ( 1st ` B ) ) -> ( x <Q u -> x e. ( 1st ` B ) ) )
6614, 15, 65syl2anc 415 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( x <Q u -> x e. ( 1st ` B ) ) )
6761, 63, 663jaod 1345 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> ( ( u <Q x \/ u = x \/ x <Q u ) -> x e. ( 1st ` B ) ) )
6827, 67mpd 13 . . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) /\ ( u e. ( 1st ` B ) /\ ( u +Q t ) e. ( 2nd ` B ) ) ) -> x e. ( 1st ` B ) )
6913, 68rexlimddv 2673 . . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) /\ ( t e. Q. /\ ( x +Q t ) = s ) ) -> x e. ( 1st ` B ) )
706, 69rexlimddv 2673 . . . . 5 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) /\ ( s e. ( 1st ` A ) /\ x <Q s ) ) -> x e. ( 1st ` B ) )
714, 70rexlimddv 2673 . . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> x e. ( 1st ` B ) )
7271expr 375 . . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ -. B <P A ) -> ( x e. ( 1st ` A ) -> x e. ( 1st ` B ) ) )
73723impa 1225 . 2 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( x e. ( 1st ` A ) -> x e. ( 1st ` B ) ) )
7473ssrdv 3254 1 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( 1st ` A ) C_ ( 1st ` B ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   <P cltp 7662
This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735
This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iltp 7837
This theorem is used by:  aptipr  8008
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