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Theorem distrlem1pru 7045
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 6999 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-imp 6931 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑦) ∧ ∈ (1st𝑧) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑦) ∧ ∈ (2nd𝑧) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
3 mulclnq 6838 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelvu 6975 . . . . 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 280 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
653impb 1135 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
7 df-iplp 6930 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑥) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑥) ∧ 𝑓 = (𝑔 +Q ))}⟩)
8 addclnq 6837 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelvu 6975 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
1093adant1 957 . . . . . . . . 9 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
1110adantr 270 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐶)𝑣 = (𝑦 +Q 𝑧)))
12 prop 6937 . . . . . . . . . . . . . . . . 17 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnqu 6944 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
1412, 13sylan 277 . . . . . . . . . . . . . . . 16 ((𝐴P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
15143ad2antl1 1101 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑥Q)
1615adantrr 463 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑥Q)
1716adantr 270 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑥Q)
18 prop 6937 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 elprnqu 6944 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
2018, 19sylan 277 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 prop 6937 . . . . . . . . . . . . . . . . . 18 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
22 elprnqu 6944 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (2nd𝐶)) → 𝑧Q)
2321, 22sylan 277 . . . . . . . . . . . . . . . . 17 ((𝐶P𝑧 ∈ (2nd𝐶)) → 𝑧Q)
2420, 23anim12i 331 . . . . . . . . . . . . . . . 16 (((𝐵P𝑦 ∈ (2nd𝐵)) ∧ (𝐶P𝑧 ∈ (2nd𝐶))) → (𝑦Q𝑧Q))
2524an4s 553 . . . . . . . . . . . . . . 15 (((𝐵P𝐶P) ∧ (𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶))) → (𝑦Q𝑧Q))
26253adantl1 1095 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶))) → (𝑦Q𝑧Q))
2726ad2ant2r 493 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑦Q𝑧Q))
28 3anass 924 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) ↔ (𝑥Q ∧ (𝑦Q𝑧Q)))
2917, 27, 28sylanbrc 408 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥Q𝑦Q𝑧Q))
30 simprr 499 . . . . . . . . . . . . 13 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
31 simpr 108 . . . . . . . . . . . . 13 (((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
3230, 31anim12i 331 . . . . . . . . . . . 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 5599 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
3433eqeq2d 2094 . . . . . . . . . . . . . 14 (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
3534biimpac 292 . . . . . . . . . . . . 13 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
36 distrnqg 6849 . . . . . . . . . . . . . 14 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
3736eqeq2d 2094 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) → (𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
3835, 37syl5ib 152 . . . . . . . . . . . 12 ((𝑥Q𝑦Q𝑧Q) → ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
3929, 32, 38sylc 61 . . . . . . . . . . 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 7034 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
41403adant3 959 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
4241ad2antrr 472 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
43 mulclpr 7034 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
44433adant2 958 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
4544ad2antrr 472 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
46 simpll 496 . . . . . . . . . . . . 13 (((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ (2nd𝐵))
472, 3genppreclu 6977 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
48473adant3 959 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
4948impl 372 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (2nd𝐴)) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
5049adantlrr 467 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
5146, 50sylan2 280 . . . . . . . . . . . 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 497 . . . . . . . . . . . . 13 (((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ (2nd𝐶))
532, 3genppreclu 6977 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
54533adant2 958 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
5554impl 372 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (2nd𝐴)) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
5655adantlrr 467 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
5752, 56sylan2 280 . . . . . . . . . . . 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 6977 . . . . . . . . . . . . 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 122 . . . . . . . . . . . 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 1171 . . . . . . . . . . 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 2159 . . . . . . . . . 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 357 . . . . . . . . 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 2489 . . . . . . . 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 148 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (2nd𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
6564exp32 357 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (2nd𝐴) → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6665com34 82 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (2nd𝐴) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6766impd 251 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (2nd𝐴) ∧ 𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
6867rexlimdvv 2489 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (2nd𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
696, 68sylbid 148 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
7069ssrdv 3016 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 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wrex 2354  wss 2984  cop 3425  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   +Q cplq 6744   ·Q cmq 6745  Pcnp 6753   +P cpp 6755   ·P cmp 6756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-imp 6931
This theorem is referenced by:  distrprg  7050
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