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Theorem addcanprleml 7555
Description: Lemma for addcanprg 7557. (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 7416 . . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnmaddl 7431 . . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 1st ` B ) ) -> E. w e. Q. ( v +Q w ) e. ( 1st ` B ) )
31, 2sylan 281 . . . . . 6 |- ( ( B e. P. /\ v e. ( 1st ` B ) ) -> E. w e. Q. ( v +Q w ) e. ( 1st ` B ) )
433ad2antl2 1150 . . . . 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 469 . . . 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 521 . . . . . 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 7351 . . . . . 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 523 . . . . . . . . . 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 274 . . . . . . . . 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 999 . . . . . . . 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 7416 . . . . . . . 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 521 . . . . . . 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 7445 . . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1613, 14, 15syl2anc 409 . . . . . 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 480 . . . . . . . . . . . 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 999 . . . . . . . . . . 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 1000 . . . . . . . . . . 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 7478 . . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2118, 19, 20syl2anc 409 . . . . . . . . . 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 7416 . . . . . . . . . 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 521 . . . . . . . . . . 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 7422 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
2724, 25, 26syl2anc 409 . . . . . . . . . 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 520 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 7422 . . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 1st ` B ) ) -> v e. Q. )
3228, 30, 31syl2anc 409 . . . . . . . . . . 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 525 . . . . . . . . . . . 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 274 . . . . . . . . . . 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 7316 . . . . . . . . . . 11 |- ( ( v e. Q. /\ w e. Q. ) -> ( v +Q w ) e. Q. )
3632, 34, 35syl2anc 409 . . . . . . . . . 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 7316 . . . . . . . . . 10 |- ( ( u e. Q. /\ ( v +Q w ) e. Q. ) -> ( u +Q ( v +Q w ) ) e. Q. )
3827, 36, 37syl2anc 409 . . . . . . . . 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 7433 . . . . . . . . 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 409 . . . . . . . 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 274 . . . . . . . . . 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 274 . . . . . . . . . 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 525 . . . . . . . . . 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 526 . . . . . . . . . . 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 480 . . . . . . . . . 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 7409 . . . . . . . . . . . 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 7316 . . . . . . . . . . . 12 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
4846, 47genpprecll 7455 . . . . . . . . . . 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 123 . . . . . . . . . 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 1229 . . . . . . . . 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 274 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 7322 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
5554adantl 275 . . . . . . . . . . . . 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 7323 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
5756adantl 275 . . . . . . . . . . . . 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 7316 . . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) e. Q. )
5958adantl 275 . . . . . . . . . . . . 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 6026 . . . . . . . . . . . 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 522 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 5858 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 7323 . . . . . . . . . . . . 13 |- ( ( u e. Q. /\ v e. Q. /\ w e. Q. ) -> ( ( u +Q v ) +Q w ) = ( u +Q ( v +Q w ) ) )
6651, 53, 64, 65syl3anc 1228 . . . . . . . . . . . 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 2202 . . . . . . . . . . 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 526 . . . . . . . . . . . 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 109 . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . 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 7456 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 2244 . . . . . . . . . 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 109 . . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( A +P. B ) = ( A +P. C ) )
7776ad3antrrr 484 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 5486 . . . . . . . . . . . 12 |- ( ( A +P. B ) = ( A +P. C ) -> ( 2nd ` ( A +P. B ) ) = ( 2nd ` ( A +P. C ) ) )
8079eleq2d 2236 . . . . . . . . . . 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 166 . . . . . . . . 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 304 . . . . . . . 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 655 . . . . . . 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 7416 . . . . . . . . 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 525 . . . . . . . . 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 7348 . . . . . . . . 9 |- ( ( v e. Q. /\ t e. Q. ) -> v <Q ( v +Q t ) )
8932, 87, 88syl2anc 409 . . . . . . . 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 7432 . . . . . . . 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 409 . . . . . . 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 1339 . . . . . 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 2588 . . . . 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 2588 . . . 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 2588 . . 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 114 . 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 3148 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 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445   C_ wss 3116   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409
This theorem is referenced by:  addcanprg  7557
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