ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  distrlem4prl Structured version   GIF version

Theorem distrlem4prl 6550
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4prl (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
Distinct variable groups:   x,y,z,f,A   x,B,y,z,f   x,𝐶,y,z,f

Proof of Theorem distrlem4prl
Dummy variables w v u g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6378 . . . . . . 7 ((w Q v Q u Q) → (w <Q v ↔ (u ·Q w) <Q (u ·Q v)))
21adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q u Q)) → (w <Q v ↔ (u ·Q w) <Q (u ·Q v)))
3 simp1 903 . . . . . . 7 ((A P B P 𝐶 P) → A P)
4 simpll 481 . . . . . . 7 (((x (1stA) y (1stB)) (f (1stA) z (1st𝐶))) → x (1stA))
5 prop 6450 . . . . . . . 8 (A P → ⟨(1stA), (2ndA)⟩ P)
6 elprnql 6456 . . . . . . . 8 ((⟨(1stA), (2ndA)⟩ P x (1stA)) → x Q)
75, 6sylan 267 . . . . . . 7 ((A P x (1stA)) → x Q)
83, 4, 7syl2an 273 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → x Q)
9 simprl 483 . . . . . . 7 (((x (1stA) y (1stB)) (f (1stA) z (1st𝐶))) → f (1stA))
10 elprnql 6456 . . . . . . . 8 ((⟨(1stA), (2ndA)⟩ P f (1stA)) → f Q)
115, 10sylan 267 . . . . . . 7 ((A P f (1stA)) → f Q)
123, 9, 11syl2an 273 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → f Q)
13 simpl2 907 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → B P)
14 simprlr 490 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → y (1stB))
15 prop 6450 . . . . . . . 8 (B P → ⟨(1stB), (2ndB)⟩ P)
16 elprnql 6456 . . . . . . . 8 ((⟨(1stB), (2ndB)⟩ P y (1stB)) → y Q)
1715, 16sylan 267 . . . . . . 7 ((B P y (1stB)) → y Q)
1813, 14, 17syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → y Q)
19 mulcomnqg 6360 . . . . . . 7 ((w Q v Q) → (w ·Q v) = (v ·Q w))
2019adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q)) → (w ·Q v) = (v ·Q w))
212, 8, 12, 18, 20caovord2d 5609 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x <Q f ↔ (x ·Q y) <Q (f ·Q y)))
22 ltanqg 6377 . . . . . . 7 ((w Q v Q u Q) → (w <Q v ↔ (u +Q w) <Q (u +Q v)))
2322adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q u Q)) → (w <Q v ↔ (u +Q w) <Q (u +Q v)))
24 mulclnq 6353 . . . . . . 7 ((x Q y Q) → (x ·Q y) Q)
258, 18, 24syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q y) Q)
26 mulclnq 6353 . . . . . . 7 ((f Q y Q) → (f ·Q y) Q)
2712, 18, 26syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q y) Q)
28 simpl3 908 . . . . . . . 8 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → 𝐶 P)
29 simprrr 492 . . . . . . . 8 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → z (1st𝐶))
30 prop 6450 . . . . . . . . 9 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
31 elprnql 6456 . . . . . . . . 9 ((⟨(1st𝐶), (2nd𝐶)⟩ P z (1st𝐶)) → z Q)
3230, 31sylan 267 . . . . . . . 8 ((𝐶 P z (1st𝐶)) → z Q)
3328, 29, 32syl2anc 391 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → z Q)
34 mulclnq 6353 . . . . . . 7 ((f Q z Q) → (f ·Q z) Q)
3512, 33, 34syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q z) Q)
36 addcomnqg 6358 . . . . . . 7 ((w Q v Q) → (w +Q v) = (v +Q w))
3736adantl 262 . . . . . 6 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) (w Q v Q)) → (w +Q v) = (v +Q w))
3823, 25, 27, 35, 37caovord2d 5609 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) <Q (f ·Q y) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
3921, 38bitrd 177 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x <Q f ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
40 simpl1 906 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → A P)
41 addclpr 6513 . . . . . . . 8 ((B P 𝐶 P) → (B +P 𝐶) P)
42413adant1 921 . . . . . . 7 ((A P B P 𝐶 P) → (B +P 𝐶) P)
4342adantr 261 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (B +P 𝐶) P)
44 mulclpr 6543 . . . . . 6 ((A P (B +P 𝐶) P) → (A ·P (B +P 𝐶)) P)
4540, 43, 44syl2anc 391 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (A ·P (B +P 𝐶)) P)
46 distrnqg 6364 . . . . . . 7 ((f Q y Q z Q) → (f ·Q (y +Q z)) = ((f ·Q y) +Q (f ·Q z)))
4712, 18, 33, 46syl3anc 1134 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q (y +Q z)) = ((f ·Q y) +Q (f ·Q z)))
48 simprrl 491 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → f (1stA))
49 df-iplp 6443 . . . . . . . . . 10 +P = (u P, v P ↦ ⟨{w Qg Q Q (g (1stu) (1stv) w = (g +Q ))}, {w Qg Q Q (g (2ndu) (2ndv) w = (g +Q ))}⟩)
50 addclnq 6352 . . . . . . . . . 10 ((g Q Q) → (g +Q ) Q)
5149, 50genpprecll 6489 . . . . . . . . 9 ((B P 𝐶 P) → ((y (1stB) z (1st𝐶)) → (y +Q z) (1st ‘(B +P 𝐶))))
5251imp 115 . . . . . . . 8 (((B P 𝐶 P) (y (1stB) z (1st𝐶))) → (y +Q z) (1st ‘(B +P 𝐶)))
5313, 28, 14, 29, 52syl22anc 1135 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (y +Q z) (1st ‘(B +P 𝐶)))
54 df-imp 6444 . . . . . . . . 9 ·P = (u P, v P ↦ ⟨{w Qg Q Q (g (1stu) (1stv) w = (g ·Q ))}, {w Qg Q Q (g (2ndu) (2ndv) w = (g ·Q ))}⟩)
55 mulclnq 6353 . . . . . . . . 9 ((g Q Q) → (g ·Q ) Q)
5654, 55genpprecll 6489 . . . . . . . 8 ((A P (B +P 𝐶) P) → ((f (1stA) (y +Q z) (1st ‘(B +P 𝐶))) → (f ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶)))))
5756imp 115 . . . . . . 7 (((A P (B +P 𝐶) P) (f (1stA) (y +Q z) (1st ‘(B +P 𝐶)))) → (f ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
5840, 43, 48, 53, 57syl22anc 1135 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
5947, 58eqeltrrd 2112 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((f ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
60 prop 6450 . . . . . 6 ((A ·P (B +P 𝐶)) P → ⟨(1st ‘(A ·P (B +P 𝐶))), (2nd ‘(A ·P (B +P 𝐶)))⟩ P)
61 prcdnql 6459 . . . . . 6 ((⟨(1st ‘(A ·P (B +P 𝐶))), (2nd ‘(A ·P (B +P 𝐶)))⟩ P ((f ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
6260, 61sylan 267 . . . . 5 (((A ·P (B +P 𝐶)) P ((f ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
6345, 59, 62syl2anc 391 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
6439, 63sylbid 139 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x <Q f → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
652, 12, 8, 33, 20caovord2d 5609 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f <Q x ↔ (f ·Q z) <Q (x ·Q z)))
66 mulclnq 6353 . . . . . . 7 ((x Q z Q) → (x ·Q z) Q)
678, 33, 66syl2anc 391 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q z) Q)
68 ltanqg 6377 . . . . . 6 (((f ·Q z) Q (x ·Q z) Q (x ·Q y) Q) → ((f ·Q z) <Q (x ·Q z) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
6935, 67, 25, 68syl3anc 1134 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((f ·Q z) <Q (x ·Q z) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
7065, 69bitrd 177 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f <Q x ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
71 distrnqg 6364 . . . . . . 7 ((x Q y Q z Q) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z)))
728, 18, 33, 71syl3anc 1134 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z)))
73 simprll 489 . . . . . . 7 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → x (1stA))
7454, 55genpprecll 6489 . . . . . . . 8 ((A P (B +P 𝐶) P) → ((x (1stA) (y +Q z) (1st ‘(B +P 𝐶))) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶)))))
7574imp 115 . . . . . . 7 (((A P (B +P 𝐶) P) (x (1stA) (y +Q z) (1st ‘(B +P 𝐶)))) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
7640, 43, 73, 53, 75syl22anc 1135 . . . . . 6 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
7772, 76eqeltrrd 2112 . . . . 5 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (x ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
78 prcdnql 6459 . . . . . 6 ((⟨(1st ‘(A ·P (B +P 𝐶))), (2nd ‘(A ·P (B +P 𝐶)))⟩ P ((x ·Q y) +Q (x ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
7960, 78sylan 267 . . . . 5 (((A ·P (B +P 𝐶)) P ((x ·Q y) +Q (x ·Q z)) (1st ‘(A ·P (B +P 𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
8045, 77, 79syl2anc 391 . . . 4 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
8170, 80sylbid 139 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (f <Q x → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
8264, 81jaod 636 . 2 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x <Q f f <Q x) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
83 ltsonq 6375 . . . . 5 <Q Or Q
84 nqtri3or 6373 . . . . 5 ((x Q f Q) → (x <Q f x = f f <Q x))
8583, 84sotritrieq 4052 . . . 4 ((x Q f Q) → (x = f ↔ ¬ (x <Q f f <Q x)))
868, 12, 85syl2anc 391 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x = f ↔ ¬ (x <Q f f <Q x)))
87 oveq1 5459 . . . . . . 7 (x = f → (x ·Q z) = (f ·Q z))
8887oveq2d 5468 . . . . . 6 (x = f → ((x ·Q y) +Q (x ·Q z)) = ((x ·Q y) +Q (f ·Q z)))
8972, 88sylan9eq 2089 . . . . 5 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) x = f) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (f ·Q z)))
9076adantr 261 . . . . 5 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) x = f) → (x ·Q (y +Q z)) (1st ‘(A ·P (B +P 𝐶))))
9189, 90eqeltrrd 2112 . . . 4 ((((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) x = f) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
9291ex 108 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (x = f → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
9386, 92sylbird 159 . 2 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → (¬ (x <Q f f <Q x) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶)))))
94 ltdcnq 6374 . . . . 5 ((x Q f Q) → DECID x <Q f)
95 ltdcnq 6374 . . . . . 6 ((f Q x Q) → DECID f <Q x)
9695ancoms 255 . . . . 5 ((x Q f Q) → DECID f <Q x)
97 dcor 842 . . . . 5 (DECID x <Q f → (DECID f <Q xDECID (x <Q f f <Q x)))
9894, 96, 97sylc 56 . . . 4 ((x Q f Q) → DECID (x <Q f f <Q x))
998, 12, 98syl2anc 391 . . 3 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → DECID (x <Q f f <Q x))
100 df-dc 742 . . 3 (DECID (x <Q f f <Q x) ↔ ((x <Q f f <Q x) ¬ (x <Q f f <Q x)))
10199, 100sylib 127 . 2 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x <Q f f <Q x) ¬ (x <Q f f <Q x)))
10282, 93, 101mpjaod 637 1 (((A P B P 𝐶 P) ((x (1stA) y (1stB)) (f (1stA) z (1st𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (1st ‘(A ·P (B +P 𝐶))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741   w3a 884   = wceq 1242   wcel 1390  cop 3369   class class class wbr 3754  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   +Q cplq 6259   ·Q cmq 6260   <Q cltq 6262  Pcnp 6268   +P cpp 6270   ·P cmp 6271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-2o 5934  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-enq0 6399  df-nq0 6400  df-0nq0 6401  df-plq0 6402  df-mq0 6403  df-inp 6441  df-iplp 6443  df-imp 6444
This theorem is referenced by:  distrlem5prl  6552
  Copyright terms: Public domain W3C validator