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Theorem addcanprlemu 7813
Description: Lemma for addcanprg 7814. (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 7673 . . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2 prnminu 7687 . . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
31, 2sylan 283 . . . . . 6 |- ( ( B e. P. /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
433ad2antl2 1184 . . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
54adantlr 477 . . . 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 531 . . . . . 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 7607 . . . . . 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 529 . . . . . . 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 7608 . . . . . . 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 7673 . . . . . . . . . . . . . 14 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13 prarloc2 7702 . . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1412, 13sylan 283 . . . . . . . . . . . . 13 |- ( ( A e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
1514adantrr 479 . . . . . . . . . . . 12 |- ( ( A e. P. /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
16153ad2antl1 1183 . . . . . . . . . . 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 477 . . . . . . . . . 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 477 . . . . . . . . 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 477 . . . . . . . 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 477 . . . . . . 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 533 . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . 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 1033 . . . . . . . . . . . 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 1034 . . . . . . . . . . . 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 7735 . . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2623, 24, 25syl2anc 411 . . . . . . . . . . 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 7673 . . . . . . . . . . 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 529 . . . . . . . . . . . . 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 7679 . . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
3229, 30, 31syl2anc 411 . . . . . . . . . . . 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 535 . . . . . . . . . . . 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 7573 . . . . . . . . . . . 12 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
3532, 33, 34syl2anc 411 . . . . . . . . . . 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 529 . . . . . . . . . . . . 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 492 . . . . . . . . . . . 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 7680 . . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ r e. ( 2nd ` B ) ) -> r e. Q. )
4036, 38, 39syl2anc 411 . . . . . . . . . . 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 7573 . . . . . . . . . . 11 |- ( ( ( u +Q t ) e. Q. /\ r e. Q. ) -> ( ( u +Q t ) +Q r ) e. Q. )
4235, 40, 41syl2anc 411 . . . . . . . . . 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 7690 . . . . . . . . . 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 411 . . . . . . . . 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 7580 . . . . . . . . . . . . . . 15 |- ( ( u e. Q. /\ t e. Q. /\ r e. Q. ) -> ( ( u +Q t ) +Q r ) = ( u +Q ( t +Q r ) ) )
4632, 33, 40, 45syl3anc 1271 . . . . . . . . . . . . . 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 7579 . . . . . . . . . . . . . . . 16 |- ( ( t e. Q. /\ r e. Q. ) -> ( t +Q r ) = ( r +Q t ) )
4847oveq2d 6023 . . . . . . . . . . . . . . 15 |- ( ( t e. Q. /\ r e. Q. ) -> ( u +Q ( t +Q r ) ) = ( u +Q ( r +Q t ) ) )
4933, 40, 48syl2anc 411 . . . . . . . . . . . . . 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 2262 . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 535 . . . . . . . . . . . . 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 110 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 7666 . . . . . . . . . . . . . . 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 7573 . . . . . . . . . . . . . . 15 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
5957, 58genpprecll 7712 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . 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 433 . . . . . . . . . . . 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 2306 . . . . . . . . . . 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 5629 . . . . . . . . . . . . 13 |- ( ( A +P. B ) = ( A +P. C ) -> ( 1st ` ( A +P. B ) ) = ( 1st ` ( A +P. C ) ) )
6463eleq2d 2299 . . . . . . . . . . . 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 501 . . . . . . . . . . 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 167 . . . . . . . . . 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 7713 . . . . . . . . . . . . . . . . . . 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 269 . . . . . . . . . . . . . . . . . 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 1041 . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 400 . . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 477 . . . . . . . . . . 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 276 . . . . . . . . . 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 306 . . . . . . . . 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 669 . . . . . . . 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 7673 . . . . . . . . . . 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 7605 . . . . . . . . . . . . . 14 |- ( ( t e. Q. /\ t e. Q. ) -> t <Q ( t +Q t ) )
8233, 33, 81syl2anc 411 . . . . . . . . . . . . 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 536 . . . . . . . . . . . . 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 4109 . . . . . . . . . . . 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 7600 . . . . . . . . . . . 12 |- ( ( t <Q w /\ r e. Q. ) -> ( r +Q t ) <Q ( r +Q w ) )
8684, 40, 85syl2anc 411 . . . . . . . . . . 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 531 . . . . . . . . . . . 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 488 . . . . . . . . . . 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 4109 . . . . . . . . . 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 7689 . . . . . . . . . 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 411 . . . . . . . . 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 734 . . . . . . . 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 1383 . . . . . . 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 2653 . . . . . 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 2653 . . . . 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 2653 . . . 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 2653 . . 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 115 . 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 3230 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 104   <-> wb 105   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   1stc1st 6290   2ndc2nd 6291   Q.cnq 7478   +Q cplq 7480   <Q cltq 7483   P.cnp 7489   +P. cpp 7491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-iplp 7666
This theorem is referenced by:  addcanprg  7814
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