Theorem addsrpr 7747

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 4660	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 7739	. . . . 5 |- ~R e. _V
3	2	ecelqsi 6592	. . . 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 4660	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 6592	. . . 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 2177	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2177	. . . 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 2177	. . 3 |- [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R
13		opeq12 3782	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6574	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2189	. . . . . . 7 |- ( ( w = A /\ v = B ) -> ( [ <. A , B >. ] ~R = [ <. w , v >. ] ~R <-> [ <. A , B >. ] ~R = [ <. A , B >. ] ~R ) )
16	15	anbi1d 465	. . . . . 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 5893	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( w +P. C ) = ( A +P. C ) )
19		simpr 110	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 5893	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( v +P. D ) = ( B +P. D ) )
21	18, 20	opeq12d 3788	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> <. ( w +P. C ) , ( v +P. D ) >. = <. ( A +P. C ) , ( B +P. D ) >. )
22	21	eceq1d 6574	. . . . . . 7 |- ( ( w = A /\ v = B ) -> [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
23	22	eqeq2d 2189	. . . . . 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 473	. . . . 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 2829	. . . 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 3782	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
27	26	eceq1d 6574	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
28	27	eqeq2d 2189	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> ( [ <. C , D >. ] ~R = [ <. u , t >. ] ~R <-> [ <. C , D >. ] ~R = [ <. C , D >. ] ~R ) )
29	28	anbi2d 464	. . . . . . 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 5894	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( w +P. u ) = ( w +P. C ) )
32		simpr 110	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> t = D )
33	32	oveq2d 5894	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( v +P. t ) = ( v +P. D ) )
34	31, 33	opeq12d 3788	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> <. ( w +P. u ) , ( v +P. t ) >. = <. ( w +P. C ) , ( v +P. D ) >. )
35	34	eceq1d 6574	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R )
36	35	eqeq2d 2189	. . . . . . 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 473	. . . . . 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 2829	. . . . 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 1882	. . . 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 409	. . 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 432	. 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 6542	. . . 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 997	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> x = [ <. A , B >. ] ~R )
45	44	eqeq1d 2186	. . . . . . 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 998	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> y = [ <. C , D >. ] ~R )
47	46	eqeq1d 2186	. . . . . . 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 473	. . . . . 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 999	. . . . . . 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 2186	. . . . . 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 473	. . . . 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 1870	. . . 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 7745	. . . 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 7730	. . . . 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 7729	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
56	55	eleq2i 2244	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
57	55	eleq2i 2244	. . . . . . . 8 |- ( y e. R. <-> y e. ( ( P. X. P. ) /. ~R ) )
58	56, 57	anbi12i 460	. . . . . . 7 |- ( ( x e. R. /\ y e. R. ) <-> ( x e. ( ( P. X. P. ) /. ~R ) /\ y e. ( ( P. X. P. ) /. ~R ) ) )
59	58	anbi1i 458	. . . . . 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 5933	. . . . 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 2198	. . . 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 5999	. . 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 1326	. 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 978   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739   <.cop 3597   X. cxp 4626  (class class class)co 5878   {coprab 5879   [cec 6536   /.cqs 6537   P.cnp 7293   +P. cpp 7295   ~R cer 7298   R.cnr 7299   +R cplr 7303

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-2o 6421  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-enq0 7426  df-nq0 7427  df-0nq0 7428  df-plq0 7429  df-mq0 7430  df-inp 7468  df-iplp 7470  df-enr 7728  df-nr 7729  df-plr 7730

This theorem is referenced by:  addclsr  7755  addcomsrg  7757  addasssrg  7758  distrsrg  7761  m1p1sr  7762  0idsr  7769  ltasrg  7772  prsradd  7788  pitonnlem2  7849