Theorem distrlem1prl 7338

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

Assertion

Ref	Expression
distrlem1prl	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( A .P. ( B +P. C ) ) ) C_ ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )

Proof of Theorem distrlem1prl

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 7293	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7225	. . . . . 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 7132	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvl 7268	. . . . 5 |- ( ( A e. P. /\ ( B +P. C ) e. P. ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) <-> E. x e. ( 1st ` A ) E. v e. ( 1st ` ( B +P. C ) ) w = ( x .Q v ) ) )
5	1, 4	sylan2 282	. . . 4 |- ( ( A e. P. /\ ( B e. P. /\ C e. P. ) ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) <-> E. x e. ( 1st ` A ) E. v e. ( 1st ` ( B +P. C ) ) w = ( x .Q v ) ) )
6	5	3impb 1160	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) <-> E. x e. ( 1st ` A ) E. v e. ( 1st ` ( B +P. C ) ) w = ( x .Q v ) ) )
7		df-iplp 7224	. . . . . . . . . . 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 7131	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvl 7268	. . . . . . . . . 10 |- ( ( B e. P. /\ C e. P. ) -> ( v e. ( 1st ` ( B +P. C ) ) <-> E. y e. ( 1st ` B ) E. z e. ( 1st ` C ) v = ( y +Q z ) ) )
10	9	3adant1 982	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( v e. ( 1st ` ( B +P. C ) ) <-> E. y e. ( 1st ` B ) E. z e. ( 1st ` C ) v = ( y +Q z ) ) )
11	10	adantr 272	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> ( v e. ( 1st ` ( B +P. C ) ) <-> E. y e. ( 1st ` B ) E. z e. ( 1st ` C ) v = ( y +Q z ) ) )
12		prop 7231	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnql 7237	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
14	12, 13	sylan 279	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
15	14	3ad2antl1 1126	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 1st ` A ) ) -> x e. Q. )
16	15	adantrr 468	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> x e. Q. )
17	16	adantr 272	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> x e. Q. )
18		prop 7231	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnql 7237	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
20	18, 19	sylan 279	. . . . . . . . . . . . . . . . 17 |- ( ( B e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
21		prop 7231	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnql 7237	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
23	21, 22	sylan 279	. . . . . . . . . . . . . . . . 17 |- ( ( C e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
24	20, 23	anim12i 334	. . . . . . . . . . . . . . . 16 |- ( ( ( B e. P. /\ y e. ( 1st ` B ) ) /\ ( C e. P. /\ z e. ( 1st ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
25	24	an4s 560	. . . . . . . . . . . . . . 15 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
26	25	3adantl1 1120	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
27	26	ad2ant2r 498	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( y e. Q. /\ z e. Q. ) )
28		3anass 949	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) <-> ( x e. Q. /\ ( y e. Q. /\ z e. Q. ) ) )
29	17, 27, 28	sylanbrc 411	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( x e. Q. /\ y e. Q. /\ z e. Q. ) )
30		simprr 504	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> w = ( x .Q v ) )
31		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) -> v = ( y +Q z ) )
32	30, 31	anim12i 334	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( w = ( x .Q v ) /\ v = ( y +Q z ) ) )
33		oveq2 5736	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2126	. . . . . . . . . . . . . 14 |- ( v = ( y +Q z ) -> ( w = ( x .Q v ) <-> w = ( x .Q ( y +Q z ) ) ) )
35	34	biimpac 294	. . . . . . . . . . . . 13 |- ( ( w = ( x .Q v ) /\ v = ( y +Q z ) ) -> w = ( x .Q ( y +Q z ) ) )
36		distrnqg 7143	. . . . . . . . . . . . . 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 2126	. . . . . . . . . . . . 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. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> w = ( ( x .Q y ) +Q ( x .Q z ) ) )
40		mulclpr 7328	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 984	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
42	41	ad2antrr 477	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( A .P. B ) e. P. )
43		mulclpr 7328	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 983	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
45	44	ad2antrr 477	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( A .P. C ) e. P. )
46		simpll 501	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) -> y e. ( 1st ` B ) )
47	2, 3	genpprecll 7270	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) -> ( x .Q y ) e. ( 1st ` ( A .P. B ) ) ) )
48	47	3adant3 984	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( x e. ( 1st ` A ) /\ y e. ( 1st ` B ) ) -> ( x .Q y ) e. ( 1st ` ( A .P. B ) ) ) )
49	48	impl 375	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 1st ` A ) ) /\ y e. ( 1st ` B ) ) -> ( x .Q y ) e. ( 1st ` ( A .P. B ) ) )
50	49	adantlrr 472	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ y e. ( 1st ` B ) ) -> ( x .Q y ) e. ( 1st ` ( A .P. B ) ) )
51	46, 50	sylan2 282	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( x .Q y ) e. ( 1st ` ( A .P. B ) ) )
52		simplr 502	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) -> z e. ( 1st ` C ) )
53	2, 3	genpprecll 7270	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ C e. P. ) -> ( ( x e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) -> ( x .Q z ) e. ( 1st ` ( A .P. C ) ) ) )
54	53	3adant2 983	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( x e. ( 1st ` A ) /\ z e. ( 1st ` C ) ) -> ( x .Q z ) e. ( 1st ` ( A .P. C ) ) ) )
55	54	impl 375	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 1st ` A ) ) /\ z e. ( 1st ` C ) ) -> ( x .Q z ) e. ( 1st ` ( A .P. C ) ) )
56	55	adantlrr 472	. . . . . . . . . . . . 13 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ z e. ( 1st ` C ) ) -> ( x .Q z ) e. ( 1st ` ( A .P. C ) ) )
57	52, 56	sylan2 282	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( x .Q z ) e. ( 1st ` ( A .P. C ) ) )
58	7, 8	genpprecll 7270	. . . . . . . . . . . . 13 |- ( ( ( A .P. B ) e. P. /\ ( A .P. C ) e. P. ) -> ( ( ( x .Q y ) e. ( 1st ` ( A .P. B ) ) /\ ( x .Q z ) e. ( 1st ` ( A .P. C ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 1st ` ( ( 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. ( 1st ` ( A .P. B ) ) /\ ( x .Q z ) e. ( 1st ` ( A .P. C ) ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
60	42, 45, 51, 57, 59	syl22anc 1200	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> ( ( x .Q y ) +Q ( x .Q z ) ) e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
61	39, 60	eqeltrd 2191	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) /\ ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )
62	61	exp32 360	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) -> ( v = ( y +Q z ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) )
63	62	rexlimdvv 2530	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> ( E. y e. ( 1st ` B ) E. z e. ( 1st ` C ) v = ( y +Q z ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
64	11, 63	sylbid 149	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> ( v e. ( 1st ` ( B +P. C ) ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
65	64	exp32 360	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 1st ` A ) -> ( w = ( x .Q v ) -> ( v e. ( 1st ` ( B +P. C ) ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) ) )
66	65	com34 83	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 1st ` A ) -> ( v e. ( 1st ` ( B +P. C ) ) -> ( w = ( x .Q v ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) ) )
67	66	impd 252	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( x e. ( 1st ` A ) /\ v e. ( 1st ` ( B +P. C ) ) ) -> ( w = ( x .Q v ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) ) )
68	67	rexlimdvv 2530	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. x e. ( 1st ` A ) E. v e. ( 1st ` ( B +P. C ) ) w = ( x .Q v ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
69	6, 68	sylbid 149	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( w e. ( 1st ` ( A .P. ( B +P. C ) ) ) -> w e. ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) ) )
70	69	ssrdv 3069	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 1st ` ( A .P. ( B +P. C ) ) ) C_ ( 1st ` ( ( A .P. B ) +P. ( A .P. C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 945   = wceq 1314   e. wcel 1463   E.wrex 2391   C_ wss 3037   <.cop 3496   ` cfv 5081  (class class class)co 5728   1stc1st 5990   2ndc2nd 5991   Q.cnq 7036   +Q cplq 7038   .Q cmq 7039   P.cnp 7047   +P. cpp 7049   .P. cmp 7050

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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-iplp 7224  df-imp 7225

This theorem is referenced by:  distrprg  7344