Theorem distrlem1pru 7524

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 7478	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B +P. C ) e. P. )
2		df-imp 7410	. . . . . 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 7317	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )
4	2, 3	genpelvu 7454	. . . . 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 284	. . . 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 1189	. . 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 7409	. . . . . . . . . . 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 7316	. . . . . . . . . . 11 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
9	7, 8	genpelvu 7454	. . . . . . . . . 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 1005	. . . . . . . . 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 274	. . . . . . . 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 7416	. . . . . . . . . . . . . . . . 17 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
13		elprnqu 7423	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
14	12, 13	sylan 281	. . . . . . . . . . . . . . . 16 |- ( ( A e. P. /\ x e. ( 2nd ` A ) ) -> x e. Q. )
15	14	3ad2antl1 1149	. . . . . . . . . . . . . . 15 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ x e. ( 2nd ` A ) ) -> x e. Q. )
16	15	adantrr 471	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> x e. Q. )
17	16	adantr 274	. . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . . . . . 18 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		elprnqu 7423	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
20	18, 19	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( B e. P. /\ y e. ( 2nd ` B ) ) -> y e. Q. )
21		prop 7416	. . . . . . . . . . . . . . . . . 18 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
22		elprnqu 7423	. . . . . . . . . . . . . . . . . 18 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
23	21, 22	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( C e. P. /\ z e. ( 2nd ` C ) ) -> z e. Q. )
24	20, 23	anim12i 336	. . . . . . . . . . . . . . . 16 |- ( ( ( B e. P. /\ y e. ( 2nd ` B ) ) /\ ( C e. P. /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
25	24	an4s 578	. . . . . . . . . . . . . . 15 |- ( ( ( B e. P. /\ C e. P. ) /\ ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) ) -> ( y e. Q. /\ z e. Q. ) )
26	25	3adantl1 1143	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 972	. . . . . . . . . . . . 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. ( 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 522	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. /\ C e. P. ) /\ ( x e. ( 2nd ` A ) /\ w = ( x .Q v ) ) ) -> w = ( x .Q v ) )
31		simpr 109	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` 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. ( 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 5850	. . . . . . . . . . . . . . 15 |- ( v = ( y +Q z ) -> ( x .Q v ) = ( x .Q ( y +Q z ) ) )
34	33	eqeq2d 2177	. . . . . . . . . . . . . 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 7328	. . . . . . . . . . . . . 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 2177	. . . . . . . . . . . . 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. ( 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 7513	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
41	40	3adant3 1007	. . . . . . . . . . . . 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. ( 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 7513	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ C e. P. ) -> ( A .P. C ) e. P. )
44	43	3adant2 1006	. . . . . . . . . . . . 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. ( 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 519	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> y e. ( 2nd ` B ) )
47	2, 3	genppreclu 7456	. . . . . . . . . . . . . . . 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 1007	. . . . . . . . . . . . . . 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 378	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 520	. . . . . . . . . . . . 13 |- ( ( ( y e. ( 2nd ` B ) /\ z e. ( 2nd ` C ) ) /\ v = ( y +Q z ) ) -> z e. ( 2nd ` C ) )
53	2, 3	genppreclu 7456	. . . . . . . . . . . . . . . 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 1006	. . . . . . . . . . . . . . 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 378	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 7456	. . . . . . . . . . . . 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 123	. . . . . . . . . . . 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 1229	. . . . . . . . . . 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 2243	. . . . . . . . . 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 363	. . . . . . . . 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 2590	. . . . . . . 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 149	. . . . . . 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 363	. . . . . 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 252	. . . 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 2590	. . 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 149	. 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 3148	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445   C_ wss 3116   <.cop 3579   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   .Q cmq 7224   P.cnp 7232   +P. cpp 7234   .P. cmp 7235

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-imp 7410

This theorem is referenced by:  distrprg  7529