Theorem addsrpr 7807

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 4692	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> <. A , B >. e. ( P. X. P. ) )
2		enrex 7799	. . . . 5 |- ~R e. _V
3	2	ecelqsi 6645	. . . 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 4692	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> <. C , D >. e. ( P. X. P. ) )
6	2	ecelqsi 6645	. . . 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 2193	. . . 4 |- [ <. A , B >. ] ~R = [ <. A , B >. ] ~R
10		eqid 2193	. . . 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 2193	. . 3 |- [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R
13		opeq12 3807	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14	13	eceq1d 6625	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> [ <. w , v >. ] ~R = [ <. A , B >. ] ~R )
15	14	eqeq2d 2205	. . . . . . 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 5934	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( w +P. C ) = ( A +P. C ) )
19		simpr 110	. . . . . . . . . 10 |- ( ( w = A /\ v = B ) -> v = B )
20	19	oveq1d 5934	. . . . . . . . 9 |- ( ( w = A /\ v = B ) -> ( v +P. D ) = ( B +P. D ) )
21	18, 20	opeq12d 3813	. . . . . . . 8 |- ( ( w = A /\ v = B ) -> <. ( w +P. C ) , ( v +P. D ) >. = <. ( A +P. C ) , ( B +P. D ) >. )
22	21	eceq1d 6625	. . . . . . 7 |- ( ( w = A /\ v = B ) -> [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R )
23	22	eqeq2d 2205	. . . . . 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 2851	. . . 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 3807	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
27	26	eceq1d 6625	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> [ <. u , t >. ] ~R = [ <. C , D >. ] ~R )
28	27	eqeq2d 2205	. . . . . . . 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 5935	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( w +P. u ) = ( w +P. C ) )
32		simpr 110	. . . . . . . . . . 11 |- ( ( u = C /\ t = D ) -> t = D )
33	32	oveq2d 5935	. . . . . . . . . 10 |- ( ( u = C /\ t = D ) -> ( v +P. t ) = ( v +P. D ) )
34	31, 33	opeq12d 3813	. . . . . . . . 9 |- ( ( u = C /\ t = D ) -> <. ( w +P. u ) , ( v +P. t ) >. = <. ( w +P. C ) , ( v +P. D ) >. )
35	34	eceq1d 6625	. . . . . . . 8 |- ( ( u = C /\ t = D ) -> [ <. ( w +P. u ) , ( v +P. t ) >. ] ~R = [ <. ( w +P. C ) , ( v +P. D ) >. ] ~R )
36	35	eqeq2d 2205	. . . . . . 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 2851	. . . . 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 1893	. . . 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 6593	. . . 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 999	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> x = [ <. A , B >. ] ~R )
45	44	eqeq1d 2202	. . . . . . 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 1000	. . . . . . . 8 |- ( ( x = [ <. A , B >. ] ~R /\ y = [ <. C , D >. ] ~R /\ z = [ <. ( A +P. C ) , ( B +P. D ) >. ] ~R ) -> y = [ <. C , D >. ] ~R )
47	46	eqeq1d 2202	. . . . . . 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 1001	. . . . . . 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 2202	. . . . . 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 1881	. . . 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 7805	. . . 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 7790	. . . . 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 7789	. . . . . . . . 9 |- R. = ( ( P. X. P. ) /. ~R )
56	55	eleq2i 2260	. . . . . . . 8 |- ( x e. R. <-> x e. ( ( P. X. P. ) /. ~R ) )
57	55	eleq2i 2260	. . . . . . . 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 5974	. . . . 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 2214	. . . 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 6041	. . 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 1337	. 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 980   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   <.cop 3622   X. cxp 4658  (class class class)co 5919   {coprab 5920   [cec 6587   /.cqs 6588   P.cnp 7353   +P. cpp 7355   ~R cer 7358   R.cnr 7359   +R cplr 7363

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iplp 7530  df-enr 7788  df-nr 7789  df-plr 7790

This theorem is referenced by:  addclsr  7815  addcomsrg  7817  addasssrg  7818  distrsrg  7821  m1p1sr  7822  0idsr  7829  ltasrg  7832  prsradd  7848  pitonnlem2  7909