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Theorem aptiprleml 7996
Description: Lemma for aptipr 7998. (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 7832 . . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 prnmaxl 7845 . . . . . . 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 7766 . . . . . . 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 7832 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
11 prarloc2 7861 . . . . . . . . 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 7838 . . . . . . . . . . 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 7838 . . . . . . . . . . 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 7753 . . . . . . . . 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 7732 . . . . . . . . . . . . . 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 7757 . . . . . . . . . . . . . . . . . 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 7738 . . . . . . . . . . . . . . . . . 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 6249 . . . . . . . . . . . . . . . 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 7841 . . . . . . . . . . . . . . . . . 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 7863 . . . . . . . . . . . . . 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 7841 . . . . . . . . . . 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
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
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