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Theorem distrlem1pru 7557
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem1pru ((𝐴P𝐵P𝐶P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Proof of Theorem distrlem1pru
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 7511 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-imp 7443 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑦) ∧ ∈ (1st𝑧) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑦) ∧ ∈ (2nd𝑧) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
3 mulclnq 7350 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelvu 7487 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
51, 4sylan2 286 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
653impb 1199 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
7 df-iplp 7442 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑥) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑥) ∧ 𝑓 = (𝑔 +Q ))}⟩)
8 addclnq 7349 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelvu 7487 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1015 . . . . . . . . 9 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
1110adantr 276 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
12 prop 7449 . . . . . . . . . . . . . . . . 17 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnqu 7456 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
1412, 13sylan 283 . . . . . . . . . . . . . . . 16 ((𝐴P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
15143ad2antl1 1159 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑥Q)
1615adantrr 479 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑥Q)
1716adantr 276 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑥Q)
18 prop 7449 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 elprnqu 7456 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
2018, 19sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 prop 7449 . . . . . . . . . . . . . . . . . 18 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
22 elprnqu 7456 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (2nd𝐶)) → 𝑧Q)
2321, 22sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐶P𝑧 ∈ (2nd𝐶)) → 𝑧Q)
2420, 23anim12i 338 . . . . . . . . . . . . . . . 16 (((𝐵P𝑦 ∈ (2nd𝐵)) ∧ (𝐶P𝑧 ∈ (2nd𝐶))) → (𝑦Q𝑧Q))
2524an4s 588 . . . . . . . . . . . . . . 15 (((𝐵P𝐶P) ∧ (𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶))) → (𝑦Q𝑧Q))
26253adantl1 1153 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶))) → (𝑦Q𝑧Q))
2726ad2ant2r 509 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑦Q𝑧Q))
28 3anass 982 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) ↔ (𝑥Q ∧ (𝑦Q𝑧Q)))
2917, 27, 28sylanbrc 417 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥Q𝑦Q𝑧Q))
30 simprr 531 . . . . . . . . . . . . 13 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
31 simpr 110 . . . . . . . . . . . . 13 (((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
3230, 31anim12i 338 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)))
33 oveq2 5873 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
3433eqeq2d 2187 . . . . . . . . . . . . . 14 (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
3534biimpac 298 . . . . . . . . . . . . 13 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
36 distrnqg 7361 . . . . . . . . . . . . . 14 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
3736eqeq2d 2187 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) → (𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
3835, 37syl5ib 154 . . . . . . . . . . . 12 ((𝑥Q𝑦Q𝑧Q) → ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
3929, 32, 38sylc 62 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
40 mulclpr 7546 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
41403adant3 1017 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
4241ad2antrr 488 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
43 mulclpr 7546 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
44433adant2 1016 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
4544ad2antrr 488 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
46 simpll 527 . . . . . . . . . . . . 13 (((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ (2nd𝐵))
472, 3genppreclu 7489 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
48473adant3 1017 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
4948impl 380 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (2nd𝐴)) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
5049adantlrr 483 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
5146, 50sylan2 286 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
52 simplr 528 . . . . . . . . . . . . 13 (((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ (2nd𝐶))
532, 3genppreclu 7489 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
54533adant2 1016 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
5554impl 380 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (2nd𝐴)) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
5655adantlrr 483 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
5752, 56sylan2 286 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
587, 8genppreclu 7489 . . . . . . . . . . . . 13 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (((𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
5958imp 124 . . . . . . . . . . . 12 ((((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) ∧ ((𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
6042, 45, 51, 57, 59syl22anc 1239 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
6139, 60eqeltrd 2252 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
6261exp32 365 . . . . . . . . 9 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
6362rexlimdvv 2599 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
6411, 63sylbid 150 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
6564exp32 365 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (2nd𝐴) → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6665com34 83 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (2nd𝐴) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6766impd 254 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
6867rexlimdvv 2599 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
696, 68sylbid 150 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
7069ssrdv 3159 1 ((𝐴P𝐵P𝐶P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2146  wrex 2454  wss 3127  cop 3592  cfv 5208  (class class class)co 5865  1st c1st 6129  2nd c2nd 6130  Qcnq 7254   +Q cplq 7256   ·Q cmq 7257  Pcnp 7265   +P cpp 7267   ·P cmp 7268
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-iplp 7442  df-imp 7443
This theorem is referenced by:  distrprg  7562
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