Theorem distrlem1pru 7914

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 7868	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7800	. . . . . 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 7707	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvu 7844	. . . . 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 286	. . . 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 1226	. . 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 7799	. . . . . . . . . . 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 7706	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvu 7844	. . . . . . . . . 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 1042	. . . . . . . . 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 276	. . . . . . . 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 7806	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnqu 7813	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
14	12, 13	sylan 283	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
15	14	3ad2antl1 1186	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 2nd ` A ) ) -> x e. Q. )
16	15	adantrr 479	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> x e. Q. )
17	16	adantr 276	. . . . . . . . . . . . 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 7806	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnqu 7813	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
20	18, 19	sylan 283	. . . . . . . . . . . . . . . . 17 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
21		prop 7806	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnqu 7813	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
23	21, 22	sylan 283	. . . . . . . . . . . . . . . . 17 |- ( ( C e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
24	20, 23	anim12i 338	. . . . . . . . . . . . . . . 16 |- ( ( ( B e. P. /\ y e. ( 2nd ` B ) ) /\ ( C e. P. /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
25	24	an4s 592	. . . . . . . . . . . . . . 15 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
26	25	3adantl1 1180	. . . . . . . . . . . . . 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 509	. . . . . . . . . . . . 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 1009	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) <-> ( x e. Q. /\ ( y e. Q. /\ z e. Q. ) ) )
29	17, 27, 28	sylanbrc 417	. . . . . . . . . . . 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 533	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> w = ( x .Q v ) )
31		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> v = ( y +Q z ) )
32	30, 31	anim12i 338	. . . . . . . . . . . 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 6066	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2246	. . . . . . . . . . . . . 14 |- ( v = ( y +Q z ) -> ( w = ( x .Q v ) <-> w = ( x .Q ( y +Q z ) ) ) )
35	34	biimpac 298	. . . . . . . . . . . . 13 |- ( ( w = ( x .Q v ) /\ v = ( y +Q z ) ) -> w = ( x .Q ( y +Q z ) ) )
36		distrnqg 7718	. . . . . . . . . . . . . 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 2246	. . . . . . . . . . . . 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	imbitrid 154	. . . . . . . . . . . 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 7903	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 1044	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
42	41	ad2antrr 488	. . . . . . . . . . . 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 7903	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 1043	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
45	44	ad2antrr 488	. . . . . . . . . . . 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 527	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> y e. ( 2nd ` B ) )
47	2, 3	genppreclu 7846	. . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . . . 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 380	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 286	. . . . . . . . . . . 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 529	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> z e. ( 2nd ` C ) )
53	2, 3	genppreclu 7846	. . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . 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 380	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 286	. . . . . . . . . . . 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 7846	. . . . . . . . . . . . 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 124	. . . . . . . . . . . 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 1275	. . . . . . . . . . 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 2311	. . . . . . . . . 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 365	. . . . . . . . 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 2669	. . . . . . . 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 150	. . . . . . 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 365	. . . . . 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 254	. . . 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 2669	. . 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 150	. 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 3248	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 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   C_ wss 3214   <.cop 3697   ` cfv 5357  (class class class)co 6058   1stc1st 6345   2ndc2nd 6346   Q.cnq 7611   +Q cplq 7613   .Q cmq 7614   P.cnp 7622   +P. cpp 7624   .P. cmp 7625

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-imp 7800

This theorem is referenced by:  distrprg  7919