Theorem distrlem1pru 7802

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 7756	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7688	. . . . . 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 7595	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvu 7732	. . . . 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 1225	. . 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 7687	. . . . . . . . . . 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 7594	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvu 7732	. . . . . . . . . 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 1041	. . . . . . . . 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 7694	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnqu 7701	. . . . . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . . . 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 7694	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnqu 7701	. . . . . . . . . . . . . . . . . 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 7694	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnqu 7701	. . . . . . . . . . . . . . . . . 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 1179	. . . . . . . . . . . . . 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 1008	. . . . . . . . . . . . 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 6025	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2243	. . . . . . . . . . . . . 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 7606	. . . . . . . . . . . . . 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 2243	. . . . . . . . . . . . 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 7791	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 1043	. . . . . . . . . . . . 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 7791	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 1042	. . . . . . . . . . . . 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 7734	. . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . 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 7734	. . . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . . . 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 7734	. . . . . . . . . . . . 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 1274	. . . . . . . . . . 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 2308	. . . . . . . . . 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 2657	. . . . . . . 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 2657	. . 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 3233	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 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   <.cop 3672   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   +Q cplq 7501   .Q cmq 7502   P.cnp 7510   +P. cpp 7512   .P. cmp 7513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-iplp 7687  df-imp 7688

This theorem is referenced by:  distrprg  7807