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Theorem addcanprleml 7429
Description: Lemma for addcanprg 7431. (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 7290 . . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnmaddl 7305 . . . . . . 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 7225 . . . . . 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 7290 . . . . . . . 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 7319 . . . . . . 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 7352 . . . . . . . . . . 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 7290 . . . . . . . . . 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 7296 . . . . . . . . . . 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 7296 . . . . . . . . . . . 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 7190 . . . . . . . . . . 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 7190 . . . . . . . . . 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 7307 . . . . . . . . 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 7283 . . . . . . . . . . . 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 7190 . . . . . . . . . . . 12 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
4846, 47genpprecll 7329 . . . . . . . . . . 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 7196 . . . . . . . . . . . . . 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 7197 . . . . . . . . . . . . . 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 7190 . . . . . . . . . . . . . 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 5955 . . . . . . . . . . . 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 5790 . . . . . . . . . . . 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 7197 . . . . . . . . . . . . 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 2176 . . . . . . . . . . 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 7330 . . . . . . . . . . . . 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 2217 . . . . . . . . . 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 5421 . . . . . . . . . . . 12 |- ( ( A +P. B ) = ( A +P. C ) -> ( 2nd ` ( A +P. B ) ) = ( 2nd ` ( A +P. C ) ) )
8079eleq2d 2209 . . . . . . . . . . 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 7290 . . . . . . . . 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 7222 . . . . . . . . 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 7306 . . . . . . . 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 2554 . . . . 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 2554 . . . 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 2554 . . 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 3103 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 2417   C_ wss 3071   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095   +Q cplq 7097   <Q cltq 7100   P.cnp 7106   +P. cpp 7108
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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283
This theorem is referenced by:  addcanprg  7431
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