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Theorem distrlem1prl 7062
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem1prl ((𝐴P𝐵P𝐶P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Proof of Theorem distrlem1prl
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 7017 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-imp 6949 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑦) ∧ ∈ (1st𝑧) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑦) ∧ ∈ (2nd𝑧) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
3 mulclnq 6856 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelvl 6992 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
51, 4sylan2 280 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
653impb 1137 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
7 df-iplp 6948 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑥) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑥) ∧ 𝑓 = (𝑔 +Q ))}⟩)
8 addclnq 6855 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelvl 6992 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧)))
1093adant1 959 . . . . . . . . 9 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧)))
1110adantr 270 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧)))
12 prop 6955 . . . . . . . . . . . . . . . . 17 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnql 6961 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
1412, 13sylan 277 . . . . . . . . . . . . . . . 16 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
15143ad2antl1 1103 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (1st𝐴)) → 𝑥Q)
1615adantrr 463 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑥Q)
1716adantr 270 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑥Q)
18 prop 6955 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 elprnql 6961 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
2018, 19sylan 277 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
21 prop 6955 . . . . . . . . . . . . . . . . . 18 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
22 elprnql 6961 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (1st𝐶)) → 𝑧Q)
2321, 22sylan 277 . . . . . . . . . . . . . . . . 17 ((𝐶P𝑧 ∈ (1st𝐶)) → 𝑧Q)
2420, 23anim12i 331 . . . . . . . . . . . . . . . 16 (((𝐵P𝑦 ∈ (1st𝐵)) ∧ (𝐶P𝑧 ∈ (1st𝐶))) → (𝑦Q𝑧Q))
2524an4s 553 . . . . . . . . . . . . . . 15 (((𝐵P𝐶P) ∧ (𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶))) → (𝑦Q𝑧Q))
26253adantl1 1097 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶))) → (𝑦Q𝑧Q))
2726ad2ant2r 493 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑦Q𝑧Q))
28 3anass 926 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) ↔ (𝑥Q ∧ (𝑦Q𝑧Q)))
2917, 27, 28sylanbrc 408 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥Q𝑦Q𝑧Q))
30 simprr 499 . . . . . . . . . . . . 13 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
31 simpr 108 . . . . . . . . . . . . 13 (((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
3230, 31anim12i 331 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)))
33 oveq2 5602 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
3433eqeq2d 2096 . . . . . . . . . . . . . 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 6867 . . . . . . . . . . . . . 14 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
3736eqeq2d 2096 . . . . . . . . . . . . 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 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
40 mulclpr 7052 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
41403adant3 961 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
4241ad2antrr 472 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
43 mulclpr 7052 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
44433adant2 960 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
4544ad2antrr 472 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
46 simpll 496 . . . . . . . . . . . . 13 (((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ (1st𝐵))
472, 3genpprecll 6994 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵))))
48473adant3 961 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵))))
4948impl 372 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (1st𝐴)) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)))
5049adantlrr 467 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)))
5146, 50sylan2 280 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)))
52 simplr 497 . . . . . . . . . . . . 13 (((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ (1st𝐶))
532, 3genpprecll 6994 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶))))
54533adant2 960 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶))))
5554impl 372 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (1st𝐴)) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶)))
5655adantlrr 467 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶)))
5752, 56sylan2 280 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶)))
587, 8genpprecll 6994 . . . . . . . . . . . . 13 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (((𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
5958imp 122 . . . . . . . . . . . 12 ((((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) ∧ ((𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
6042, 45, 51, 57, 59syl22anc 1173 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
6139, 60eqeltrd 2161 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
6261exp32 357 . . . . . . . . 9 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
6362rexlimdvv 2491 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
6411, 63sylbid 148 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
6564exp32 357 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st𝐴) → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6665com34 82 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st𝐴) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6766impd 251 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑣 ∈ (1st ‘(𝐵 +P 𝐶))) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
6867rexlimdvv 2491 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
696, 68sylbid 148 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
7069ssrdv 3018 1 ((𝐴P𝐵P𝐶P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wcel 1436  wrex 2356  wss 2986  cop 3428  cfv 4972  (class class class)co 5594  1st c1st 5847  2nd c2nd 5848  Qcnq 6760   +Q cplq 6762   ·Q cmq 6763  Pcnp 6771   +P cpp 6773   ·P cmp 6774
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946  df-iplp 6948  df-imp 6949
This theorem is referenced by:  distrprg  7068
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