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Theorem addcanprleml 7415
Description: Lemma for addcanprg 7417. (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 7276 . . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnmaddl 7291 . . . . . . 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 1144 . . . . 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 468 . . . 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 520 . . . . . 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 7211 . . . . . 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 522 . . . . . . . . . 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 993 . . . . . . . 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 7276 . . . . . . . 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 520 . . . . . . 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 7305 . . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1613, 14, 15syl2anc 408 . . . . . 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 479 . . . . . . . . . . . 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 993 . . . . . . . . . . 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 994 . . . . . . . . . . 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 7338 . . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2118, 19, 20syl2anc 408 . . . . . . . . . 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 7276 . . . . . . . . . 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 520 . . . . . . . . . . 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 7282 . . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
2724, 25, 26syl2anc 408 . . . . . . . . . 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 519 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 7282 . . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 1st ` B ) ) -> v e. Q. )
3228, 30, 31syl2anc 408 . . . . . . . . . . 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 524 . . . . . . . . . . . 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 7176 . . . . . . . . . . 11 |- ( ( v e. Q. /\ w e. Q. ) -> ( v +Q w ) e. Q. )
3632, 34, 35syl2anc 408 . . . . . . . . . 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 7176 . . . . . . . . . 10 |- ( ( u e. Q. /\ ( v +Q w ) e. Q. ) -> ( u +Q ( v +Q w ) ) e. Q. )
3827, 36, 37syl2anc 408 . . . . . . . . 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 7293 . . . . . . . . 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 408 . . . . . . . 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 524 . . . . . . . . . 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 525 . . . . . . . . . . 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 479 . . . . . . . . . 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 7269 . . . . . . . . . . . 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 7176 . . . . . . . . . . . 12 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
4846, 47genpprecll 7315 . . . . . . . . . . 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 1217 . . . . . . . . 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 479 . . . . . . . . . . . . 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 7182 . . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . . 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 7176 . . . . . . . . . . . . . 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 5948 . . . . . . . . . . . 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 521 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 5783 . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . 13 |- ( ( u e. Q. /\ v e. Q. /\ w e. Q. ) -> ( ( u +Q v ) +Q w ) = ( u +Q ( v +Q w ) ) )
6651, 53, 64, 65syl3anc 1216 . . . . . . . . . . . 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 2174 . . . . . . . . . . 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 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 ) ) /\ ( 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 995 . . . . . . . . . . . . . 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 7316 . . . . . . . . . . . . 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 408 . . . . . . . . . . . 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 429 . . . . . . . . . . 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 2215 . . . . . . . . . 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 483 . . . . . . . . . . . 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 479 . . . . . . . . . . 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 5414 . . . . . . . . . . . 12 |- ( ( A +P. B ) = ( A +P. C ) -> ( 2nd ` ( A +P. B ) ) = ( 2nd ` ( A +P. C ) ) )
8079eleq2d 2207 . . . . . . . . . . 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 654 . . . . . . 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 7276 . . . . . . . . 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 524 . . . . . . . . 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 7208 . . . . . . . . 9 |- ( ( v e. Q. /\ t e. Q. ) -> v <Q ( v +Q t ) )
8932, 87, 88syl2anc 408 . . . . . . . 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 7292 . . . . . . . 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 408 . . . . . . 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 1327 . . . . . 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 2552 . . . . 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 2552 . . . 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 2552 . . 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 3098 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 697   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   C_ wss 3066   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   +Q cplq 7083   <Q cltq 7086   P.cnp 7092   +P. cpp 7094
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269
This theorem is referenced by:  addcanprg  7417
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