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Theorem aptiprleml 7656
Description: Lemma for aptipr 7658. (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 7492 . . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 prnmaxl 7505 . . . . . . 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 7426 . . . . . . 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 7492 . . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
11 prarloc2 7521 . . . . . . . . 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 7498 . . . . . . . . . . 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 7498 . . . . . . . . . . 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 7413 . . . . . . . . 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 7392 . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . 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 7398 . . . . . . . . . . . . . . . . . 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 6061 . . . . . . . . . . . . . . . 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 2266 . . . . . . . . . . . . . . . . 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 7501 . . . . . . . . . . . . . . . . . 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 2252 . . . . . . . . . . . . . . 15 |- ( v = ( u +Q t ) -> ( v e. ( 2nd ` B ) <-> ( u +Q t ) e. ( 2nd ` B ) ) )
50 eleq1 2252 . . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . 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 7523 . . . . . . . . . . . . . 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 2252 . . . . . . . . . 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 7501 . . . . . . . . . . 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 1315 . . . . . . . 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 2612 . . . . . 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 2612 . . . . 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 2612 . . . 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 3176 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 1364   e. wcel 2160   E.wrex 2469   C_ wss 3144   <.cop 3610   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   1stc1st 6157   2ndc2nd 6158   Q.cnq 7297   +Q cplq 7299   <Q cltq 7302   P.cnp 7308   <P cltp 7312
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-pli 7322  df-mi 7323  df-lti 7324  df-plpq 7361  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-plqqs 7366  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-enq0 7441  df-nq0 7442  df-0nq0 7443  df-plq0 7444  df-mq0 7445  df-inp 7483  df-iltp 7487
This theorem is referenced by:  aptipr  7658
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