Theorem distrlem1prl 7514

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 7469	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7401	. . . . . 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 7308	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvl 7444	. . . . 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 284	. . . 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 1188	. . 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 7400	. . . . . . . . . . 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 7307	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvl 7444	. . . . . . . . . 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 1004	. . . . . . . . 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 274	. . . . . . . 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 7407	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnql 7413	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
14	12, 13	sylan 281	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> x e. Q. )
15	14	3ad2antl1 1148	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 1st ` A ) ) -> x e. Q. )
16	15	adantrr 471	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 1st ` A ) /\ w = ( x .Q v ) ) ) -> x e. Q. )
17	16	adantr 274	. . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnql 7413	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
20	18, 19	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( B e. P. /\ y e. ( 1st ` B ) ) -> y e. Q. )
21		prop 7407	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnql 7413	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
23	21, 22	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( C e. P. /\ z e. ( 1st ` C ) ) -> z e. Q. )
24	20, 23	anim12i 336	. . . . . . . . . . . . . . . 16 |- ( ( ( B e. P. /\ y e. ( 1st ` B ) ) /\ ( C e. P. /\ z e. ( 1st ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
25	24	an4s 578	. . . . . . . . . . . . . . 15 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
26	25	3adantl1 1142	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 971	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) <-> ( x e. Q. /\ ( y e. Q. /\ z e. Q. ) ) )
29	17, 27, 28	sylanbrc 414	. . . . . . . . . . . 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 522	. . . . . . . . . . . . 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 336	. . . . . . . . . . . 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 5844	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2176	. . . . . . . . . . . . . 14 |- ( v = ( y +Q z ) -> ( w = ( x .Q v ) <-> w = ( x .Q ( y +Q z ) ) ) )
35	34	biimpac 296	. . . . . . . . . . . . 13 |- ( ( w = ( x .Q v ) /\ v = ( y +Q z ) ) -> w = ( x .Q ( y +Q z ) ) )
36		distrnqg 7319	. . . . . . . . . . . . . 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 2176	. . . . . . . . . . . . 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 7504	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 1006	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. B ) e. P. )
42	41	ad2antrr 480	. . . . . . . . . . . 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 7504	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 1005	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
45	44	ad2antrr 480	. . . . . . . . . . . 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 519	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) -> y e. ( 1st ` B ) )
47	2, 3	genpprecll 7446	. . . . . . . . . . . . . . . 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 1006	. . . . . . . . . . . . . . 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 378	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 520	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 1st ` B ) /\ z e. ( 1st ` C ) ) /\ v = ( y +Q z ) ) -> z e. ( 1st ` C ) )
53	2, 3	genpprecll 7446	. . . . . . . . . . . . . . . 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 1005	. . . . . . . . . . . . . . 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 378	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 7446	. . . . . . . . . . . . 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 1228	. . . . . . . . . . 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 2241	. . . . . . . . . 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 363	. . . . . . . . 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 2588	. . . . . . . 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 363	. . . . . 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 2588	. . 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 3143	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 967   = wceq 1342   e. wcel 2135   E.wrex 2443   C_ wss 3111   <.cop 3573   ` cfv 5182  (class class class)co 5836   1stc1st 6098   2ndc2nd 6099   Q.cnq 7212   +Q cplq 7214   .Q cmq 7215   P.cnp 7223   +P. cpp 7225   .P. cmp 7226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-2o 6376  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-rq 7284  df-ltnqqs 7285  df-enq0 7356  df-nq0 7357  df-0nq0 7358  df-plq0 7359  df-mq0 7360  df-inp 7398  df-iplp 7400  df-imp 7401

This theorem is referenced by:  distrprg  7520