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Theorem distrlem1prl 7642
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 7597 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-imp 7529 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑦) ∧ ∈ (1st𝑧) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑦) ∧ ∈ (2nd𝑧) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
3 mulclnq 7436 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelvl 7572 . . . . 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 286 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
653impb 1201 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
7 df-iplp 7528 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑥) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑥) ∧ 𝑓 = (𝑔 +Q ))}⟩)
8 addclnq 7435 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelvl 7572 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1017 . . . . . . . . 9 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧)))
1110adantr 276 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐶)𝑣 = (𝑦 +Q 𝑧)))
12 prop 7535 . . . . . . . . . . . . . . . . 17 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
13 elprnql 7541 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
1412, 13sylan 283 . . . . . . . . . . . . . . . 16 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
15143ad2antl1 1161 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (1st𝐴)) → 𝑥Q)
1615adantrr 479 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑥Q)
1716adantr 276 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑥Q)
18 prop 7535 . . . . . . . . . . . . . . . . . 18 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 elprnql 7541 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
2018, 19sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
21 prop 7535 . . . . . . . . . . . . . . . . . 18 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
22 elprnql 7541 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (1st𝐶)) → 𝑧Q)
2321, 22sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐶P𝑧 ∈ (1st𝐶)) → 𝑧Q)
2420, 23anim12i 338 . . . . . . . . . . . . . . . 16 (((𝐵P𝑦 ∈ (1st𝐵)) ∧ (𝐶P𝑧 ∈ (1st𝐶))) → (𝑦Q𝑧Q))
2524an4s 588 . . . . . . . . . . . . . . 15 (((𝐵P𝐶P) ∧ (𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶))) → (𝑦Q𝑧Q))
26253adantl1 1155 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ (𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶))) → (𝑦Q𝑧Q))
2726ad2ant2r 509 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑦Q𝑧Q))
28 3anass 984 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q𝑧Q) ↔ (𝑥Q ∧ (𝑦Q𝑧Q)))
2917, 27, 28sylanbrc 417 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥Q𝑦Q𝑧Q))
30 simprr 531 . . . . . . . . . . . . 13 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
31 simpr 110 . . . . . . . . . . . . 13 (((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
3230, 31anim12i 338 . . . . . . . . . . . 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 5926 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
3433eqeq2d 2205 . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . 14 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
3736eqeq2d 2205 . . . . . . . . . . . . 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 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
40 mulclpr 7632 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
41403adant3 1019 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
4241ad2antrr 488 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
43 mulclpr 7632 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
44433adant2 1018 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
4544ad2antrr 488 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
46 simpll 527 . . . . . . . . . . . . 13 (((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ (1st𝐵))
472, 3genpprecll 7574 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵))))
48473adant3 1019 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵))))
4948impl 380 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (1st𝐴)) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)))
5049adantlrr 483 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦 ∈ (1st𝐵)) → (𝑥 ·Q 𝑦) ∈ (1st ‘(𝐴 ·P 𝐵)))
5146, 50sylan2 286 . . . . . . . . . . . 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 528 . . . . . . . . . . . . 13 (((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ (1st𝐶))
532, 3genpprecll 7574 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶))))
54533adant2 1018 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶))))
5554impl 380 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥 ∈ (1st𝐴)) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶)))
5655adantlrr 483 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧 ∈ (1st𝐶)) → (𝑥 ·Q 𝑧) ∈ (1st ‘(𝐴 ·P 𝐶)))
5752, 56sylan2 286 . . . . . . . . . . . 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 7574 . . . . . . . . . . . . 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 124 . . . . . . . . . . . 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 1250 . . . . . . . . . . 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 2270 . . . . . . . . . 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 365 . . . . . . . . 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 2618 . . . . . . . 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 150 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥 ∈ (1st𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
6564exp32 365 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st𝐴) → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6665com34 83 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥 ∈ (1st𝐴) → (𝑣 ∈ (1st ‘(𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
6766impd 254 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥 ∈ (1st𝐴) ∧ 𝑣 ∈ (1st ‘(𝐵 +P 𝐶))) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
6867rexlimdvv 2618 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (1st𝐴)∃𝑣 ∈ (1st ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
696, 68sylbid 150 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) → 𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
7069ssrdv 3185 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 104  wb 105  w3a 980   = wceq 1364  wcel 2164  wrex 2473  wss 3153  cop 3621  cfv 5254  (class class class)co 5918  1st c1st 6191  2nd c2nd 6192  Qcnq 7340   +Q cplq 7342   ·Q cmq 7343  Pcnp 7351   +P cpp 7353   ·P cmp 7354
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-imp 7529
This theorem is referenced by:  distrprg  7648
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