Theorem distrlem1pru 7667

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 7621	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7553	. . . . . 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 7460	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvu 7597	. . . . 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 1201	. . 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 7552	. . . . . . . . . . 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 7459	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvu 7597	. . . . . . . . . 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 1017	. . . . . . . . 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 7559	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnqu 7566	. . . . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . . 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 7559	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnqu 7566	. . . . . . . . . . . . . . . . . 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 7559	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnqu 7566	. . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 15 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
26	25	3adantl1 1155	. . . . . . . . . . . . . 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 984	. . . . . . . . . . . . 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 531	. . . . . . . . . . . . 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 5933	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2208	. . . . . . . . . . . . . 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 7471	. . . . . . . . . . . . . 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 2208	. . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 1019	. . . . . . . . . . . . 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 7656	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 1018	. . . . . . . . . . . . 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 7599	. . . . . . . . . . . . . . . 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 1019	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> z e. ( 2nd ` C ) )
53	2, 3	genppreclu 7599	. . . . . . . . . . . . . . . 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 1018	. . . . . . . . . . . . . . 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 7599	. . . . . . . . . . . . 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 1250	. . . . . . . . . . 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 2273	. . . . . . . . . 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 2621	. . . . . . . 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 2621	. . 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 3190	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 980   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   <.cop 3626   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   +Q cplq 7366   .Q cmq 7367   P.cnp 7375   +P. cpp 7377   .P. cmp 7378

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-imp 7553

This theorem is referenced by:  distrprg  7672