Theorem addcanprlemu 7727

Description: Lemma for addcanprg 7728. (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.

Step	Hyp	Ref	Expression
1		prop 7587	. . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prnminu 7601	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
3	1, 2	sylan 283	. . . . . 6 |- ( ( B e. P. /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
4	3	3ad2antl2 1162	. . . . 5 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ v e. ( 2nd ` B ) ) -> E. r e. ( 2nd ` B ) r <Q v )
5	4	adantlr 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 7521	. . . . . 6 |- ( r <Q v -> E. w e. Q. ( r +Q w ) = v )
8	6, 7	syl 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 7522	. . . . . . 7 |- ( w e. Q. -> E. t e. Q. ( t +Q t ) = w )
11	9, 10	syl 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 7587	. . . . . . . . . . . . . 14 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		prarloc2 7616	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
14	12, 13	sylan 283	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ t e. Q. ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
15	14	adantrr 479	. . . . . . . . . . . 12 |- ( ( A e. P. /\ ( t e. Q. /\ ( t +Q t ) = w ) ) -> E. u e. ( 1st ` A ) ( u +Q t ) e. ( 2nd ` A ) )
16	15	3ad2antl1 1161	. . . . . . . . . . 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 ) )
17	16	adantlr 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 ) )
18	17	adantlr 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 ) )
19	18	adantlr 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 ) )
20	19	adantlr 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. ) )
22	21	ad3antrrr 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. ) )
23	22	simp1d 1011	. . . . . . . . . . . 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. )
24	22	simp2d 1012	. . . . . . . . . . . 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 7649	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
26	23, 24, 25	syl2anc 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 7587	. . . . . . . . . . 11 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
28	26, 27	syl 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. )
29	23, 12	syl 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 7593	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
32	29, 30, 31	syl2anc 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 7487	. . . . . . . . . . . 12 |- ( ( u e. Q. /\ t e. Q. ) -> ( u +Q t ) e. Q. )
35	32, 33, 34	syl2anc 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. )
36	24, 1	syl 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 ) )
38	37	ad3antrrr 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 7594	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ r e. ( 2nd ` B ) ) -> r e. Q. )
40	36, 38, 39	syl2anc 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 7487	. . . . . . . . . . 11 |- ( ( ( u +Q t ) e. Q. /\ r e. Q. ) -> ( ( u +Q t ) +Q r ) e. Q. )
42	35, 40, 41	syl2anc 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 7604	. . . . . . . . . 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 ) ) ) )
44	28, 42, 43	syl2anc 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 7494	. . . . . . . . . . . . . . 15 |- ( ( u e. Q. /\ t e. Q. /\ r e. Q. ) -> ( ( u +Q t ) +Q r ) = ( u +Q ( t +Q r ) ) )
46	32, 33, 40, 45	syl3anc 1249	. . . . . . . . . . . . . 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 7493	. . . . . . . . . . . . . . . 16 |- ( ( t e. Q. /\ r e. Q. ) -> ( t +Q r ) = ( r +Q t ) )
48	47	oveq2d 5959	. . . . . . . . . . . . . . 15 |- ( ( t e. Q. /\ r e. Q. ) -> ( u +Q ( t +Q r ) ) = ( u +Q ( r +Q t ) ) )
49	33, 40, 48	syl2anc 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 ) ) )
50	46, 49	eqtrd 2237	. . . . . . . . . . . . 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 ) ) )
51	50	adantr 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 ) )
54	23	adantr 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. )
55	22	simp3d 1013	. . . . . . . . . . . . . . 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. )
56	55	adantr 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 7580	. . . . . . . . . . . . . . 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 7487	. . . . . . . . . . . . . . 15 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
59	57, 58	genpprecll 7626	. . . . . . . . . . . . . 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 ) ) ) )
60	54, 56, 59	syl2anc 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 ) ) ) )
61	52, 53, 60	mp2and 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 ) ) )
62	51, 61	eqeltrd 2281	. . . . . . . . . . 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 5575	. . . . . . . . . . . . 13 |- ( ( A +P. B ) = ( A +P. C ) -> ( 1st ` ( A +P. B ) ) = ( 1st ` ( A +P. C ) ) )
64	63	eleq2d 2274	. . . . . . . . . . . 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 ) ) ) )
65	64	ad7antlr 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 ) ) ) )
66	62, 65	mpbird 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 ) ) )
67	57, 58	genppreclu 7627	. . . . . . . . . . . . . . . . . . 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 ) ) ) )
68	67	ancomsd 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 ) ) ) )
69	68	3adant3 1019	. . . . . . . . . . . . . . . . 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 ) ) ) )
70	69	ad2antrr 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 ) ) ) )
71	70	imp 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 ) ) )
72	71	adantrlr 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 ) ) )
73	72	anassrs 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 ) ) )
74	73	ad2ant2rl 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 ) ) )
75	74	adantlr 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 ) ) )
76	75	adantr 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 ) ) )
77	66, 76	jca 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 ) ) ) )
78	44, 77	mtand 666	. . . . . . . 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 7587	. . . . . . . . . . 11 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
80	55, 79	syl 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 7519	. . . . . . . . . . . . . 14 |- ( ( t e. Q. /\ t e. Q. ) -> t <Q ( t +Q t ) )
82	33, 33, 81	syl2anc 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 )
84	82, 83	breqtrd 4069	. . . . . . . . . . . 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 7514	. . . . . . . . . . . 12 |- ( ( t <Q w /\ r e. Q. ) -> ( r +Q t ) <Q ( r +Q w ) )
86	84, 40, 85	syl2anc 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 )
88	87	ad2antrr 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 )
89	86, 88	breqtrd 4069	. . . . . . . . . 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 7603	. . . . . . . . . 10 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ ( r +Q t ) <Q v ) -> ( ( r +Q t ) e. ( 1st ` C ) \/ v e. ( 2nd ` C ) ) )
91	80, 89, 90	syl2anc 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 ) ) )
92	91	orcomd 730	. . . . . . . 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 ) ) )
93	78, 92	ecased 1361	. . . . . . 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 ) )
94	20, 93	rexlimddv 2627	. . . . . 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 ) )
95	11, 94	rexlimddv 2627	. . . . 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 ) )
96	8, 95	rexlimddv 2627	. . . 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 ) )
97	5, 96	rexlimddv 2627	. . 3 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) /\ v e. ( 2nd ` B ) ) -> v e. ( 2nd ` C ) )
98	97	ex 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 ) ) )
99	98	ssrdv 3198	1 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( A +P. B ) = ( A +P. C ) ) -> ( 2nd ` B ) C_ ( 2nd ` C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1372   e. wcel 2175   E.wrex 2484   C_ wss 3165   <.cop 3635   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   1stc1st 6223   2ndc2nd 6224   Q.cnq 7392   +Q cplq 7394   <Q cltq 7397   P.cnp 7403   +P. cpp 7405

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-iplp 7580

This theorem is referenced by:  addcanprg  7728