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Theorem addcanprleml 7613
Description: Lemma for addcanprg 7615. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprleml |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
Proof of Theorem addcanprleml
Dummy variables f g h r s t u v w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7474 . . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnmaddl 7489 . . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 1st ` B ) ) -> E. w e. Q. ( v +Q w ) e. ( 1st ` B ) )
31, 2sylan 283 . . . . . 6 |- ( ( B e. P. /\ v e. ( 1st ` B ) ) -> E. w e. Q. ( v +Q w ) e. ( 1st ` B ) )
433ad2antl2 1160 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ v e. ( 1st ` B ) ) -> E. w e. Q. ( v +Q w ) e. ( 1st ` B ) )
54adantlr 477 . . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) -> E. w e. Q. ( v +Q w ) e. ( 1st ` B ) )
6 simprl 529 . . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) -> w e. Q. )
7 halfnqq 7409 . . . . . 6 |- ( w e. Q. -> E. t e. Q. ( t +Q t ) = w )
86, 7syl 14 . . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) -> E. t e. Q. ( t +Q t ) = w )
9 simplll 533 . . . . . . . . . 10 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) -> ( A e. P. /\ B e. P. /\ C e. P. ) )
109adantr 276 . . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> ( A e. P. /\ B e. P. /\ C e. P. ) )
1110simp1d 1009 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> A e. P. )
12 prop 7474 . . . . . . . 8 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
1311, 12syl 14 . . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
14 simprl 529 . . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> t e. Q. )
15 prarloc2 7503 . . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1613, 14, 15syl2anc 411 . . . . . 6 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
179ad2antrr 488 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( A e. P. /\ B e. P. /\ C e. P. ) )
1817simp1d 1009 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> A e. P. )
1917simp2d 1010 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> B e. P. )
20 addclpr 7536 . . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2118, 19, 20syl2anc 411 . . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( A +P. B ) e. P. )
22 prop 7474 . . . . . . . . . 10 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
2321, 22syl 14 . . . . . . . . 9 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
2418, 12syl 14 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
25 simprl 529 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> u e. ( 1st ` A ) )
26 elprnql 7480 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
2724, 25, 26syl2anc 411 . . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> u e. Q. )
2819, 1syl 14 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
29 simplr 528 . . . . . . . . . . . . 13 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) -> v e. ( 1st ` B ) )
3029ad2antrr 488 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> v e. ( 1st ` B ) )
31 elprnql 7480 . . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 1st ` B ) ) -> v e. Q. )
3228, 30, 31syl2anc 411 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> v e. Q. )
33 simplrl 535 . . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> w e. Q. )
3433adantr 276 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> w e. Q. )
35 addclnq 7374 . . . . . . . . . . 11 |- ( ( v e. Q. /\ w e. Q. ) -> ( v +Q w ) e. Q. )
3632, 34, 35syl2anc 411 . . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( v +Q w ) e. Q. )
37 addclnq 7374 . . . . . . . . . 10 |- ( ( u e. Q. /\ ( v +Q w ) e. Q. ) -> ( u +Q ( v +Q w ) ) e. Q. )
3827, 36, 37syl2anc 411 . . . . . . . . 9 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( u +Q ( v +Q w ) ) e. Q. )
39 prdisj 7491 . . . . . . . . 9 |- ( ( <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. /\ ( u +Q ( v +Q w ) ) e. Q. ) -> -. ( ( u +Q ( v +Q w ) ) e. ( 1st ` ( A +P. B ) ) /\ ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. B ) ) ) )
4023, 38, 39syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> -. ( ( u +Q ( v +Q w ) ) e. ( 1st ` ( A +P. B ) ) /\ ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. B ) ) ) )
4118adantr 276 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> A e. P. )
4219adantr 276 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> B e. P. )
43 simplrl 535 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> u e. ( 1st ` A ) )
44 simplrr 536 . . . . . . . . . . 11 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> ( v +Q w ) e. ( 1st ` B ) )
4544ad2antrr 488 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( v +Q w ) e. ( 1st ` B ) )
46 df-iplp 7467 . . . . . . . . . . . 12 |- +P. = ( r e. P. , s e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` r ) /\ h e. ( 1st ` s ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` r ) /\ h e. ( 2nd ` s ) /\ f = ( g +Q h ) ) } >. )
47 addclnq 7374 . . . . . . . . . . . 12 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
4846, 47genpprecll 7513 . . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( ( u e. ( 1st ` A ) /\ ( v +Q w ) e. ( 1st ` B ) ) -> ( u +Q ( v +Q w ) ) e. ( 1st ` ( A +P. B ) ) ) )
4948imp 124 . . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( u e. ( 1st ` A ) /\ ( v +Q w ) e. ( 1st ` B ) ) ) -> ( u +Q ( v +Q w ) ) e. ( 1st ` ( A +P. B ) ) )
5041, 42, 43, 45, 49syl22anc 1239 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( u +Q ( v +Q w ) ) e. ( 1st ` ( A +P. B ) ) )
5127adantr 276 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> u e. Q. )
5214ad2antrr 488 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> t e. Q. )
5332adantr 276 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> v e. Q. )
54 addcomnqg 7380 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
5554adantl 277 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
56 addassnqg 7381 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
5756adantl 277 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
58 addclnq 7374 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) e. Q. )
5958adantl 277 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) e. Q. )
6051, 52, 53, 55, 57, 52, 59caov4d 6059 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q t ) +Q ( v +Q t ) ) = ( ( u +Q v ) +Q ( t +Q t ) ) )
61 simprr 531 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> ( t +Q t ) = w )
6261ad2antrr 488 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( t +Q t ) = w )
6362oveq2d 5891 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q v ) +Q ( t +Q t ) ) = ( ( u +Q v ) +Q w ) )
6433ad2antrr 488 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> w e. Q. )
65 addassnqg 7381 . . . . . . . . . . . . 13 |- ( ( u e. Q. /\ v e. Q. /\ w e. Q. ) -> ( ( u +Q v ) +Q w ) = ( u +Q ( v +Q w ) ) )
6651, 53, 64, 65syl3anc 1238 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q v ) +Q w ) = ( u +Q ( v +Q w ) ) )
6760, 63, 663eqtrd 2214 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q t ) +Q ( v +Q t ) ) = ( u +Q ( v +Q w ) ) )
68 simplrr 536 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( u +Q t ) e. ( 2nd ` A ) )
69 simpr 110 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( v +Q t ) e. ( 2nd ` C ) )
7017simp3d 1011 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> C e. P. )
7170adantr 276 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> C e. P. )
7246, 47genppreclu 7514 . . . . . . . . . . . . 13 |- ( ( A e. P. /\ C e. P. ) -> ( ( ( u +Q t ) e. ( 2nd ` A ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q t ) +Q ( v +Q t ) ) e. ( 2nd ` ( A +P. C ) ) ) )
7341, 71, 72syl2anc 411 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( ( u +Q t ) e. ( 2nd ` A ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q t ) +Q ( v +Q t ) ) e. ( 2nd ` ( A +P. C ) ) ) )
7468, 69, 73mp2and 433 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q t ) +Q ( v +Q t ) ) e. ( 2nd ` ( A +P. C ) ) )
7567, 74eqeltrrd 2255 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. C ) ) )
76 simpr 110 . . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( A +P. B ) = ( A +P. C ) )
7776ad3antrrr 492 . . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> ( A +P. B ) = ( A +P. C ) )
7877ad2antrr 488 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( A +P. B ) = ( A +P. C ) )
79 fveq2 5516 . . . . . . . . . . . 12 |- ( ( A +P. B ) = ( A +P. C ) -> ( 2nd ` ( A +P. B ) ) = ( 2nd ` ( A +P. C ) ) )
8079eleq2d 2247 . . . . . . . . . . 11 |- ( ( A +P. B ) = ( A +P. C ) -> ( ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. B ) ) <-> ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. C ) ) ) )
8178, 80syl 14 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. B ) ) <-> ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. C ) ) ) )
8275, 81mpbird 167 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. B ) ) )
8350, 82jca 306 . . . . . . . 8 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( v +Q t ) e. ( 2nd ` C ) ) -> ( ( u +Q ( v +Q w ) ) e. ( 1st ` ( A +P. B ) ) /\ ( u +Q ( v +Q w ) ) e. ( 2nd ` ( A +P. B ) ) ) )
8440, 83mtand 665 . . . . . . 7 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> -. ( v +Q t ) e. ( 2nd ` C ) )
85 prop 7474 . . . . . . . . 9 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
8670, 85syl 14 . . . . . . . 8 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
87 simplrl 535 . . . . . . . . 9 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> t e. Q. )
88 ltaddnq 7406 . . . . . . . . 9 |- ( ( v e. Q. /\ t e. Q. ) -> v <Q ( v +Q t ) )
8932, 87, 88syl2anc 411 . . . . . . . 8 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> v <Q ( v +Q t ) )
90 prloc 7490 . . . . . . . 8 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ v <Q ( v +Q t ) ) -> ( v e. ( 1st ` C ) \/ ( v +Q t ) e. ( 2nd ` C ) ) )
9186, 89, 90syl2anc 411 . . . . . . 7 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( v e. ( 1st ` C ) \/ ( v +Q t ) e. ( 2nd ` C ) ) )
9284, 91ecased 1349 . . . . . 6 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> v e. ( 1st ` C ) )
9316, 92rexlimddv 2599 . . . . 5 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> v e. ( 1st ` C ) )
948, 93rexlimddv 2599 . . . 4 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) /\ ( w e. Q. /\ ( v +Q w ) e. ( 1st ` B ) ) ) -> v e. ( 1st ` C ) )
955, 94rexlimddv 2599 . . 3 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 1st ` B ) ) -> v e. ( 1st ` C ) )
9695ex 115 . 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( v e. ( 1st ` B ) -> v e. ( 1st ` C ) ) )
9796ssrdv 3162 1 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 1st ` B ) C_ ( 1st ` C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ 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. cpp 7292
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-iplp 7467
This theorem is referenced by:  addcanprg  7615
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