Theorem addsrpr 8102

Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
addsrpr	|- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )

Proof of Theorem addsrpr

Dummy variables x y z w v u t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxpi 4801	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 8094	. . . . 5 |- ~R e. _V
3	2	ecelqsi 6853	. . . 4 |- ( <. A , B >. e. ( P. X. P. ) -> [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
4	1, 3	syl 14	. . 3 |- ( ( A e. P. /\ B e. P. ) -> [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
5		opelxpi 4801	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 6853	. . . 4 |- ( <. C , D >. e. ( P. X. P. ) -> [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
7	5, 6	syl 14	. . 3 |- ( ( C e. P. /\ D e. P. ) -> [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
8	4, 7	anim12i 338	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) )
9		eqid 2238	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2238	. . . 4 |- [ <. C , D >. ] ~R = [ <. C , D >. ] ~R
11	9, 10	pm3.2i 272	. . 3 |- ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R )
12		eqid 2238	. . 3 |- [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R
13		opeq12 3901	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6833	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2250	. . . . . . 7 |- ( ( w = A /\ v = B ) -> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. A , B >. ] ~R ) )
16	15	anbi1d 469	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) ) )
17		simpl 109	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> w = A )
18	17	oveq1d 6090	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( w +P. C ) = ( A +P. C ) )
19		simpr 110	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 6090	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( v +P. D ) = ( B +P. D ) )
21	18, 20	opeq12d 3907	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> <. ( w +P. C ) , ( v +P. D ) >. = <. ( A +P. C ) , ( B +P. D ) >. )
22	21	eceq1d 6833	. . . . . . 7 |- ( ( w = A /\ v = B ) -> [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
23	22	eqeq2d 2250	. . . . . 6 |- ( ( w = A /\ v = B ) -> ( [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R <-> [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) )
24	16, 23	anbi12d 477	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) ) )
25	24	spc2egv 2915	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> E. w E. v ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R ) ) )
26		opeq12 3901	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
27	26	eceq1d 6833	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
28	27	eqeq2d 2250	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> ( [ <. C , D >. ] ~R = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) )
29	28	anbi2d 468	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) ) )
30		simpl 109	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> u = C )
31	30	oveq2d 6091	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( w +P. u ) = ( w +P. C ) )
32		simpr 110	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> t = D )
33	32	oveq2d 6091	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( v +P. t ) = ( v +P. D ) )
34	31, 33	opeq12d 3907	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> <. ( w +P. u ) , ( v +P. t ) >. = <. ( w +P. C ) , ( v +P. D ) >. )
35	34	eceq1d 6833	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R )
36	35	eqeq2d 2250	. . . . . . 7 |- ( ( u = C /\ t = D ) -> ( [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R <-> [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R ) )
37	29, 36	anbi12d 477	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R ) ) )
38	37	spc2egv 2915	. . . . 5 |- ( ( C e. P. /\ D e. P. ) -> ( ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R ) -> E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
39	38	2eximdv 1935	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> ( E. w E. v ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
40	25, 39	sylan9 413	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( ( [ <. A , B >. ] ~R = [ <. A , B >. ] ~R /\ [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
41	11, 12, 40	mp2ani 436	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
42		ecexg 6801	. . . 4 |- ( ~R e. _V -> [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R e. _V )
43	2, 42	ax-mp 5	. . 3 |- [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R e. _V
44		simp1 1028	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> x = [ <. A , B >. ] ~R )
45	44	eqeq1d 2247	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> ( x = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. w , v >. ] ~R ) )
46		simp2 1029	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> y = [ <. C , D >. ] ~R )
47	46	eqeq1d 2247	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> ( y = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) )
48	45, 47	anbi12d 477	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) <-> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) ) )
49		simp3 1030	. . . . . . 7 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
50	49	eqeq1d 2247	. . . . . 6 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> ( z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R <-> [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
51	48, 50	anbi12d 477	. . . . 5 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> ( ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
52	51	4exbidv 1923	. . . 4 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> ( E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) <-> E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
53		addsrmo 8100	. . . 4 |- ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) )
54		df-plr 8085	. . . . 5 |- +R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) }
55		df-nr 8084	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
56	55	eleq2i 2305	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
57	55	eleq2i 2305	. . . . . . . 8 |- ( y e. R. <-> y e. ( ( P. X. P. ) /. ~R ) )
58	56, 57	anbi12i 464	. . . . . . 7 |- ( ( x e. R. /\ y e. R. ) <-> ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) )
59	58	anbi1i 462	. . . . . 6 |- ( ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) <-> ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) )
60	59	oprabbii 6133	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) }
61	54, 60	eqtri 2259	. . . 4 |- +R = { <. <. x , y >. , z >. | ( ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , t >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) ) }
62	52, 53, 61	ovig 6200	. . 3 |- ( ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R e. _V ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) )
63	43, 62	mp3an3 1367	. 2 |- ( ( [ <. A , B >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ [ <. C , D >. ] ~R e. ( ( P. X. P. ) /. ~R ) ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R /\ [ <. C , D >. ] ~R = [ <. u , t >. ] ~R ) /\ [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) )
64	8, 41, 63	sylc 62	1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( [ <. A , B >. ] ~R +R [ <. C , D >. ] ~R ) = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3708   X. cxp 4767  (class class class)co 6075   {coprab 6076   [cec 6795   /.cqs 6796   P.cnp 7648   +P. cpp 7650   ~R cer 7653   R.cnr 7654   +R cplr 7658

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-enr 8083  df-nr 8084  df-plr 8085

This theorem is referenced by:  addclsr  8110  addcomsrg  8112  addasssrg  8113  distrsrg  8116  m1p1sr  8117  0idsr  8124  ltasrg  8127  prsradd  8143  pitonnlem2  8204