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Theorem distrlem1pru 7617
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 7571 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-imp 7503 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑦) ∧ ∈ (1st𝑧) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑦) ∧ ∈ (2nd𝑧) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
3 mulclnq 7410 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelvu 7547 . . . . 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 1201 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
7 df-iplp 7502 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑥) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑥) ∧ 𝑓 = (𝑔 +Q ))}⟩)
8 addclnq 7409 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelvu 7547 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1017 . . . . . . . . 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 7509 . . . . . . . . . . . . . . . . 17 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnqu 7516 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
1412, 13sylan 283 . . . . . . . . . . . . . . . 16 ((𝐴P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
15143ad2antl1 1161 . . . . . . . . . . . . . . 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 7509 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 elprnqu 7516 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
2018, 19sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 prop 7509 . . . . . . . . . . . . . . . . . 18 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
22 elprnqu 7516 . . . . . . . . . . . . . . . . . 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 1155 . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . 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 5908 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
3433eqeq2d 2201 . . . . . . . . . . . . . 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 7421 . . . . . . . . . . . . . 14 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
3736eqeq2d 2201 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) → (𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
3835, 37imbitrid 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 7606 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
41403adant3 1019 . . . . . . . . . . . . 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 7606 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
44433adant2 1018 . . . . . . . . . . . . 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 7549 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
48473adant3 1019 . . . . . . . . . . . . . . 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 7549 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
54533adant2 1018 . . . . . . . . . . . . . . 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 7549 . . . . . . . . . . . . 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 1250 . . . . . . . . . . 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 2266 . . . . . . . . . 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 2614 . . . . . . . 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 2614 . . 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 3176 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 980   = wceq 1364  wcel 2160  wrex 2469  wss 3144  cop 3613  cfv 5238  (class class class)co 5900  1st c1st 6167  2nd c2nd 6168  Qcnq 7314   +Q cplq 7316   ·Q cmq 7317  Pcnp 7325   +P cpp 7327   ·P cmp 7328
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-eprel 4310  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-irdg 6399  df-1o 6445  df-2o 6446  df-oadd 6449  df-omul 6450  df-er 6563  df-ec 6565  df-qs 6569  df-ni 7338  df-pli 7339  df-mi 7340  df-lti 7341  df-plpq 7378  df-mpq 7379  df-enq 7381  df-nqqs 7382  df-plqqs 7383  df-mqqs 7384  df-1nqqs 7385  df-rq 7386  df-ltnqqs 7387  df-enq0 7458  df-nq0 7459  df-0nq0 7460  df-plq0 7461  df-mq0 7462  df-inp 7500  df-iplp 7502  df-imp 7503
This theorem is referenced by:  distrprg  7622
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