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Theorem aptiprleml 7751
Description: Lemma for aptipr 7753. (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 7587 . . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 prnmaxl 7600 . . . . . . 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 511 . . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> E. s e. ( 1st ` A ) x <Q s )
5 ltexnqi 7521 . . . . . . 7 |- ( x <Q s -> E. t e. Q. ( x +Q t ) = s )
65ad2antll 491 . . . . . 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 528 . . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> B e. P. )
87ad2antrr 488 . . . . . . . 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 529 . . . . . . . 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 7587 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
11 prarloc2 7616 . . . . . . . . 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 411 . . . . . . 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 529 . . . . . . . . . 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 7593 . . . . . . . . . . 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 411 . . . . . . . . 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 527 . . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> A e. P. )
2019ad3antrrr 492 . . . . . . . . . 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 531 . . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( -. B <P A /\ x e. ( 1st ` A ) ) ) -> x e. ( 1st ` A ) )
2221ad3antrrr 492 . . . . . . . . . 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 7593 . . . . . . . . . . 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 411 . . . . . . . . 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 7508 . . . . . . . . 9 |- ( ( u e. Q. /\ x e. Q. ) -> ( u <Q x \/ u = x \/ x <Q u ) )
2718, 25, 26syl2anc 411 . . . . . . . 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 535 . . . . . . . . . . . . . . 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 7487 . . . . . . . . . . . . . 14 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
3228, 30, 31syl2anc 411 . . . . . . . . . . . . 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 7512 . . . . . . . . . . . . . . . . . 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 7493 . . . . . . . . . . . . . . . . . 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 6115 . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . 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 2281 . . . . . . . . . . . . . . . . 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 7596 . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 531 . . . . . . . . . . . . . . 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 2267 . . . . . . . . . . . . . . 15 |- ( v = ( u +Q t ) -> ( v e. ( 2nd ` B ) <-> ( u +Q t ) e. ( 2nd ` B ) ) )
50 eleq1 2267 . . . . . . . . . . . . . . 15 |- ( v = ( u +Q t ) -> ( v e. ( 1st ` A ) <-> ( u +Q t ) e. ( 1st ` A ) ) )
5149, 50anbi12d 473 . . . . . . . . . . . . . 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 2876 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 7618 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 535 . . . . . . . . . . . 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 492 . . . . . . . . . . 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 621 . . . . . . . . . 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 2267 . . . . . . . . . 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 7596 . . . . . . . . . . 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 411 . . . . . . . . 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 1316 . . . . . . . 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 2627 . . . . . 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 2627 . . . . 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 2627 . . . 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 1196 . 2 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( x e. ( 1st ` A ) -> x e. ( 1st ` B ) ) )
7473ssrdv 3198 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 979   /\ w3a 980   = wceq 1372   e. wcel 2175   E.wrex 2484   C_ wss 3165   <.cop 3635   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   1stc1st 6223   2ndc2nd 6224   Q.cnq 7392   +Q cplq 7394   <Q cltq 7397   P.cnp 7403   <P cltp 7407
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-iltp 7582
This theorem is referenced by:  aptipr  7753
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