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Theorem aptiprleml 7638
Description: Lemma for aptipr 7640. (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 7474 . . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 prnmaxl 7487 . . . . . . 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 7408 . . . . . . 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 7474 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
11 prarloc2 7503 . . . . . . . . 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 7480 . . . . . . . . . . 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 7480 . . . . . . . . . . 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 7395 . . . . . . . . 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 7374 . . . . . . . . . . . . . 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 7399 . . . . . . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . . . . . . 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 6044 . . . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . . . . . . 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 7483 . . . . . . . . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . 15 |- ( v = ( u +Q t ) -> ( v e. ( 2nd ` B ) <-> ( u +Q t ) e. ( 2nd ` B ) ) )
50 eleq1 2240 . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . 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 7505 . . . . . . . . . . . . . 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 620 . . . . . . . . . 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 2240 . . . . . . . . . 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 7483 . . . . . . . . . . 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 1304 . . . . . . . 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 2599 . . . . . 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 2599 . . . . 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 2599 . . . 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 1194 . 2 |- ( ( A e. P. /\ B e. P. /\ -. B <P A ) -> ( x e. ( 1st ` A ) -> x e. ( 1st ` B ) ) )
7473ssrdv 3162 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 977   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   C_ wss 3130   <.cop 3596   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   +Q cplq 7281   <Q cltq 7284   P.cnp 7290   <P cltp 7294
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-iltp 7469
This theorem is referenced by:  aptipr  7640
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