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Theorem addcanprlemu 7577
Description: Lemma for addcanprg 7578. (Contributed by Jim Kingdon, 25-Dec-2019.)
Assertion
Ref Expression
addcanprlemu |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
Proof of Theorem addcanprlemu
Dummy variables f g h q r s t u v w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7437 . . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnminu 7451 . . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
31, 2sylan 281 . . . . . 6 |- ( ( B e. P. /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
433ad2antl2 1155 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
54adantlr 474 . . . 4 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
6 simprr 527 . . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) -> r <Q v )
7 ltexnqi 7371 . . . . . 6 |- ( r <Q v -> E. w e. Q. ( r +Q w ) = v )
86, 7syl 14 . . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) -> E. w e. Q. ( r +Q w ) = v )
9 simprl 526 . . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) -> w e. Q. )
10 halfnqq 7372 . . . . . . 7 |- ( w e. Q. -> E. t e. Q. ( t +Q t ) = w )
119, 10syl 14 . . . . . 6 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) -> E. t e. Q. ( t +Q t ) = w )
12 prop 7437 . . . . . . . . . . . . . 14 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13 prarloc2 7466 . . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1412, 13sylan 281 . . . . . . . . . . . . 13 |- ( ( A e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1514adantrr 476 . . . . . . . . . . . 12 |- ( ( A e. P. /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
16153ad2antl1 1154 . . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1716adantlr 474 . . . . . . . . . 10 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1817adantlr 474 . . . . . . . . 9 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1918adantlr 474 . . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
2019adantlr 474 . . . . . . 7 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
21 simplll 528 . . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) -> ( A e. P. /\ B e. P. /\ C e. P. ) )
2221ad3antrrr 489 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( 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. ) )
2322simp1d 1004 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> A e. P. )
2422simp2d 1005 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> B e. P. )
25 addclpr 7499 . . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2623, 24, 25syl2anc 409 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( A +P. B ) e. P. )
27 prop 7437 . . . . . . . . . . 11 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
2826, 27syl 14 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( 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. )
2923, 12syl 14 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
30 simprl 526 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> u e. ( 1st ` A ) )
31 elprnql 7443 . . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
3229, 30, 31syl2anc 409 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> u e. Q. )
33 simplrl 530 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> t e. Q. )
34 addclnq 7337 . . . . . . . . . . . 12 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
3532, 33, 34syl2anc 409 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( u +Q t ) e. Q. )
3624, 1syl 14 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
37 simprl 526 . . . . . . . . . . . . 13 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) -> r e. ( 2nd ` B ) )
3837ad3antrrr 489 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> r e. ( 2nd ` B ) )
39 elprnqu 7444 . . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ r e. ( 2nd ` B ) ) -> r e. Q. )
4036, 38, 39syl2anc 409 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> r e. Q. )
41 addclnq 7337 . . . . . . . . . . 11 |- ( ( ( u +Q t ) e. Q. /\ r e. Q. ) -> ( ( u +Q t ) +Q r ) e. Q. )
4235, 40, 41syl2anc 409 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) e. Q. )
43 prdisj 7454 . . . . . . . . . 10 |- ( ( <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. /\ ( ( u +Q t ) +Q r ) e. Q. ) -> -. ( ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. B ) ) /\ ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
4428, 42, 43syl2anc 409 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> -. ( ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. B ) ) /\ ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
45 addassnqg 7344 . . . . . . . . . . . . . . 15 |- ( ( u e. Q. /\ t e. Q. /\ r e. Q. ) -> ( ( u +Q t ) +Q r ) = ( u +Q ( t +Q r ) ) )
4632, 33, 40, 45syl3anc 1233 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) = ( u +Q ( t +Q r ) ) )
47 addcomnqg 7343 . . . . . . . . . . . . . . . 16 |- ( ( t e. Q. /\ r e. Q. ) -> ( t +Q r ) = ( r +Q t ) )
4847oveq2d 5869 . . . . . . . . . . . . . . 15 |- ( ( t e. Q. /\ r e. Q. ) -> ( u +Q ( t +Q r ) ) = ( u +Q ( r +Q t ) ) )
4933, 40, 48syl2anc 409 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( u +Q ( t +Q r ) ) = ( u +Q ( r +Q t ) ) )
5046, 49eqtrd 2203 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) = ( u +Q ( r +Q t ) ) )
5150adantr 274 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( u +Q t ) +Q r ) = ( u +Q ( r +Q t ) ) )
52 simplrl 530 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> u e. ( 1st ` A ) )
53 simpr 109 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( r +Q t ) e. ( 1st ` C ) )
5423adantr 274 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> A e. P. )
5522simp3d 1006 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> C e. P. )
5655adantr 274 . . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> C e. P. )
57 df-iplp 7430 . . . . . . . . . . . . . . 15 |- +P. = ( q e. P. , s e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` q ) /\ h e. ( 1st ` s ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` q ) /\ h e. ( 2nd ` s ) /\ f = ( g +Q h ) ) } >. )
58 addclnq 7337 . . . . . . . . . . . . . . 15 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5957, 58genpprecll 7476 . . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( ( u e. ( 1st ` A ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( u +Q ( r +Q t ) ) e. ( 1st ` ( A +P. C ) ) ) )
6054, 56, 59syl2anc 409 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( u e. ( 1st ` A ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( u +Q ( r +Q t ) ) e. ( 1st ` ( A +P. C ) ) ) )
6152, 53, 60mp2and 431 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( u +Q ( r +Q t ) ) e. ( 1st ` ( A +P. C ) ) )
6251, 61eqeltrd 2247 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. C ) ) )
63 fveq2 5496 . . . . . . . . . . . . 13 |- ( ( A +P. B ) = ( A +P. C ) -> ( 1st ` ( A +P. B ) ) = ( 1st ` ( A +P. C ) ) )
6463eleq2d 2240 . . . . . . . . . . . 12 |- ( ( A +P. B ) = ( A +P. C ) -> ( ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. B ) ) <-> ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. C ) ) ) )
6564ad7antlr 498 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. B ) ) <-> ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. C ) ) ) )
6662, 65mpbird 166 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. B ) ) )
6757, 58genppreclu 7477 . . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( u +Q t ) e. ( 2nd ` A ) /\ r e. ( 2nd ` B ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
6867ancomsd 267 . . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ B e. P. ) -> ( ( r e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 2nd ` A ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
69683adant3 1012 . . . . . . . . . . . . . . . . 17 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( r e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 2nd ` A ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
7069ad2antrr 485 . . . . . . . . . . . . . . . 16 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) -> ( ( r e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 2nd ` A ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
7170imp 123 . . . . . . . . . . . . . . 15 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) )
7271adantrlr 482 . . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( ( r e. ( 2nd ` B ) /\ r <Q v ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) )
7372anassrs 398 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( u +Q t ) e. ( 2nd ` A ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) )
7473ad2ant2rl 508 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) )
7574adantlr 474 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) )
7675adantr 274 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) )
7766, 76jca 304 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) /\ ( r +Q t ) e. ( 1st ` C ) ) -> ( ( ( u +Q t ) +Q r ) e. ( 1st ` ( A +P. B ) ) /\ ( ( u +Q t ) +Q r ) e. ( 2nd ` ( A +P. B ) ) ) )
7844, 77mtand 660 . . . . . . . 8 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> -. ( r +Q t ) e. ( 1st ` C ) )
79 prop 7437 . . . . . . . . . . 11 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
8055, 79syl 14 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
81 ltaddnq 7369 . . . . . . . . . . . . . 14 |- ( ( t e. Q. /\ t e. Q. ) -> t <Q ( t +Q t ) )
8233, 33, 81syl2anc 409 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> t <Q ( t +Q t ) )
83 simplrr 531 . . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( t +Q t ) = w )
8482, 83breqtrd 4015 . . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> t <Q w )
85 ltanqi 7364 . . . . . . . . . . . 12 |- ( ( t <Q w /\ r e. Q. ) -> ( r +Q t ) <Q ( r +Q w ) )
8684, 40, 85syl2anc 409 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( r +Q t ) <Q ( r +Q w ) )
87 simprr 527 . . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) -> ( r +Q w ) = v )
8887ad2antrr 485 . . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( r +Q w ) = v )
8986, 88breqtrd 4015 . . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( r +Q t ) <Q v )
90 prloc 7453 . . . . . . . . . 10 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ ( r +Q t ) <Q v ) -> ( ( r +Q t ) e. ( 1st ` C ) \/ v e. ( 2nd ` C ) ) )
9180, 89, 90syl2anc 409 . . . . . . . . 9 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( ( r +Q t ) e. ( 1st ` C ) \/ v e. ( 2nd ` C ) ) )
9291orcomd 724 . . . . . . . 8 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> ( v e. ( 2nd ` C ) \/ ( r +Q t ) e. ( 1st ` C ) ) )
9378, 92ecased 1344 . . . . . . 7 |- ( ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) /\ ( u e. ( 1st ` A ) /\ ( u +Q t ) e. ( 2nd ` A ) ) ) -> v e. ( 2nd ` C ) )
9420, 93rexlimddv 2592 . . . . . 6 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> v e. ( 2nd ` C ) )
9511, 94rexlimddv 2592 . . . . 5 |- ( ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) /\ ( w e. Q. /\ ( r +Q w ) = v ) ) -> v e. ( 2nd ` C ) )
968, 95rexlimddv 2592 . . . 4 |- ( ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) /\ ( r e. ( 2nd ` B ) /\ r <Q v ) ) -> v e. ( 2nd ` C ) )
975, 96rexlimddv 2592 . . 3 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) -> v e. ( 2nd ` C ) )
9897ex 114 . 2 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( v e. ( 2nd ` B ) -> v e. ( 2nd ` C ) ) )
9998ssrdv 3153 1 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   C_ wss 3121   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253   +P. cpp 7255
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430
This theorem is referenced by:  addcanprg  7578
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