Theorem distrlem1pru 7239

Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)

Assertion

Ref	Expression
distrlem1pru	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( A .P. ( B +P. C ) ) ) C_ ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )

Proof of Theorem distrlem1pru

Dummy variables x y z w v f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addclpr 7193	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7125	. . . . . 6 |- .P. = ( y e. P. , z e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` y ) /\ h e. ( 1st ` z ) /\ f = ( g .Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` y ) /\ h e. ( 2nd ` z ) /\ f = ( g .Q h ) ) } >. )
3		mulclnq 7032	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvu 7169	. . . . 5 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) <-> E. x e. ( 2nd ` A ) E. v e. ( 2nd ` ( B +P. C ) ) w = ( x .Q v ) ) )
5	1, 4	sylan2 281	. . . 4 |- ( ( A e. P. /\ ( B e. P. /\ C e. P. ) ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) <-> E. x e. ( 2nd ` A ) E. v e. ( 2nd ` ( B +P. C ) ) w = ( x .Q v ) ) )
6	5	3impb 1142	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) <-> E. x e. ( 2nd ` A ) E. v e. ( 2nd ` ( B +P. C ) ) w = ( x .Q v ) ) )
7		df-iplp 7124	. . . . . . . . . . 11 |- +P. = ( w e. P. , x e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` w ) /\ h e. ( 1st ` x ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` w ) /\ h e. ( 2nd ` x ) /\ f = ( g +Q h ) ) } >. )
8		addclnq 7031	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvu 7169	. . . . . . . . . 10 |- ( ( B e. P. /\ C e. P. ) -> ( v e. ( 2nd ` ( B +P. C ) ) <-> E. y e. ( 2nd ` B ) E. z e. ( 2nd ` C ) v = ( y +Q z ) ) )
10	9	3adant1 964	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( 2nd ` ( B +P. C ) ) <-> E. y e. ( 2nd ` B ) E. z e. ( 2nd ` C ) v = ( y +Q z ) ) )
11	10	adantr 271	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> ( v e. ( 2nd ` ( B +P. C ) ) <-> E. y e. ( 2nd ` B ) E. z e. ( 2nd ` C ) v = ( y +Q z ) ) )
12		prop 7131	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnqu 7138	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
14	12, 13	sylan 278	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
15	14	3ad2antl1 1108	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 2nd ` A ) ) -> x e. Q. )
16	15	adantrr 464	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> x e. Q. )
17	16	adantr 271	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> x e. Q. )
18		prop 7131	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnqu 7138	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
20	18, 19	sylan 278	. . . . . . . . . . . . . . . . 17 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
21		prop 7131	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnqu 7138	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
23	21, 22	sylan 278	. . . . . . . . . . . . . . . . 17 |- ( ( C e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
24	20, 23	anim12i 332	. . . . . . . . . . . . . . . 16 |- ( ( ( B e. P. /\ y e. ( 2nd ` B ) ) /\ ( C e. P. /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
25	24	an4s 556	. . . . . . . . . . . . . . 15 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
26	25	3adantl1 1102	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
27	26	ad2ant2r 494	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( y e. Q. /\ z e. Q. ) )
28		3anass 931	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) <-> ( x e. Q. /\ ( y e. Q. /\ z e. Q. ) ) )
29	17, 27, 28	sylanbrc 409	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( x e. Q. /\ y e. Q. /\ z e. Q. ) )
30		simprr 500	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> w = ( x .Q v ) )
31		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> v = ( y +Q z ) )
32	30, 31	anim12i 332	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( w = ( x .Q v ) /\ v = ( y +Q z ) ) )
33		oveq2 5698	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2106	. . . . . . . . . . . . . 14 |- ( v = ( y +Q z ) -> ( w = ( x .Q v ) <-> w = ( x .Q ( y +Q z ) ) ) )
35	34	biimpac 293	. . . . . . . . . . . . 13 |- ( ( w = ( x .Q v ) /\ v = ( y +Q z ) ) -> w = ( x .Q ( y +Q z ) ) )
36		distrnqg 7043	. . . . . . . . . . . . . 14 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x .Q ( y +Q z ) ) = ( ( x .Q y ) +Q ( x .Q z ) ) )
37	36	eqeq2d 2106	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( w = ( x .Q ( y +Q z ) ) <-> w = ( ( x .Q y ) +Q ( x .Q z ) ) ) )
38	35, 37	syl5ib 153	. . . . . . . . . . . 12 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( ( w = ( x .Q v ) /\ v = ( y +Q z ) ) -> w = ( ( x .Q y ) +Q ( x .Q z ) ) ) )
39	29, 32, 38	sylc 62	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> w = ( ( x .Q y ) +Q ( x .Q z ) ) )
40		mulclpr 7228	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 966	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
42	41	ad2antrr 473	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( A .P. B ) e. P. )
43		mulclpr 7228	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 965	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
45	44	ad2antrr 473	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( A .P. C ) e. P. )
46		simpll 497	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> y e. ( 2nd ` B ) )
47	2, 3	genppreclu 7171	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) -> ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) ) )
48	47	3adant3 966	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( x e. ( 2nd ` A ) /\ y e. ( 2nd ` B ) ) -> ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) ) )
49	48	impl 373	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 2nd ` A ) ) /\ y e. ( 2nd ` B ) ) -> ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) )
50	49	adantlrr 468	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ y e. ( 2nd ` B ) ) -> ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) )
51	46, 50	sylan2 281	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) )
52		simplr 498	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> z e. ( 2nd ` C ) )
53	2, 3	genppreclu 7171	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ C e. P. ) -> ( ( x e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) -> ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) ) )
54	53	3adant2 965	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( x e. ( 2nd ` A ) /\ z e. ( 2nd ` C ) ) -> ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) ) )
55	54	impl 373	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 2nd ` A ) ) /\ z e. ( 2nd ` C ) ) -> ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) )
56	55	adantlrr 468	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ z e. ( 2nd ` C ) ) -> ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) )
57	52, 56	sylan2 281	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) )
58	7, 8	genppreclu 7171	. . . . . . . . . . . . 13 |- ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) -> ( ( ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) /\ ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
59	58	imp 123	. . . . . . . . . . . 12 |- ( ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) /\ ( ( x .Q y ) e. ( 2nd ` ( A .P. B ) ) /\ ( x .Q z ) e. ( 2nd ` ( A .P. C ) ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
60	42, 45, 51, 57, 59	syl22anc 1182	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
61	39, 60	eqeltrd 2171	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
62	61	exp32 358	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) -> ( v = ( y +Q z ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) )
63	62	rexlimdvv 2509	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> ( E. y e. ( 2nd ` B ) E. z e. ( 2nd ` C ) v = ( y +Q z ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
64	11, 63	sylbid 149	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> ( v e. ( 2nd ` ( B +P. C ) ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
65	64	exp32 358	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 2nd ` A ) -> ( w = ( x .Q v ) -> ( v e. ( 2nd ` ( B +P. C ) ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) ) )
66	65	com34 83	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 2nd ` A ) -> ( v e. ( 2nd ` ( B +P. C ) ) -> ( w = ( x .Q v ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) ) )
67	66	impd 252	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( x e. ( 2nd ` A ) /\ v e. ( 2nd ` ( B +P. C ) ) ) -> ( w = ( x .Q v ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) )
68	67	rexlimdvv 2509	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. x e. ( 2nd ` A ) E. v e. ( 2nd ` ( B +P. C ) ) w = ( x .Q v ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
69	6, 68	sylbid 149	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 2nd ` ( A .P. ( B +P. C ) ) ) -> w e. ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
70	69	ssrdv 3045	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( A .P. ( B +P. C ) ) ) C_ ( 2nd ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 927   = wceq 1296   e. wcel 1445   E.wrex 2371   C_ wss 3013   <.cop 3469   ` cfv 5049  (class class class)co 5690   1stc1st 5947   2ndc2nd 5948   Q.cnq 6936   +Q cplq 6938   .Q cmq 6939   P.cnp 6947   +P. cpp 6949   .P. cmp 6950

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-iplp 7124  df-imp 7125

This theorem is referenced by:  distrprg  7244