P. 397 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39601-39700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	jm2.20nn 39601	Lemma 2.20 of [JonesMatijasevic] p. 696, the "first step down lemma". (Contributed by Stefan O'Rear, 27-Sep-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((𝐴 Yrm 𝑁)↑2) ∥ (𝐴 Yrm 𝑀) ↔ (𝑁 · (𝐴 Yrm 𝑁)) ∥ 𝑀))
Theorem	jm2.25lem1 39602	Lemma for jm2.26 39606. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ (𝐴 ∥ (𝐶 − 𝐷) ∨ 𝐴 ∥ (𝐶 − -𝐷))) → ((𝐴 ∥ (𝐷 − 𝐵) ∨ 𝐴 ∥ (𝐷 − -𝐵)) ↔ (𝐴 ∥ (𝐶 − 𝐵) ∨ 𝐴 ∥ (𝐶 − -𝐵))))
Theorem	jm2.25 39603	Lemma for jm2.26 39606. Remainders mod X(2n) are negaperiodic mod 2n. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐼 ∈ ℤ) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm (𝑀 + (𝐼 · (2 · 𝑁)))) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm (𝑀 + (𝐼 · (2 · 𝑁)))) − -(𝐴 Yrm 𝑀))))
Theorem	jm2.26a 39604	Lemma for jm2.26 39606. Reverse direction is required to prove forward direction, so do it separately. Induction on difference between K and M, together with the addition formula fact that adding 2N only inverts sign. (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀)) → ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀)))))
Theorem	jm2.26lem3 39605	Lemma for jm2.26 39606. Use acongrep 39584 to find K', M' ~ K, M in [ 0,N ]. Thus Y(K') ~ Y(M') and both are small; K' = M' on pain of contradicting 2.24, so K ~ M. (Contributed by Stefan O'Rear, 3-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ (0...𝑁) ∧ 𝑀 ∈ (0...𝑁)) ∧ ((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀)))) → 𝐾 = 𝑀)
Theorem	jm2.26 39606	Lemma 2.26 of [JonesMatijasevic] p. 697, the "second step down lemma". (Contributed by Stefan O'Rear, 2-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − (𝐴 Yrm 𝑀)) ∨ (𝐴 Xrm 𝑁) ∥ ((𝐴 Yrm 𝐾) − -(𝐴 Yrm 𝑀))) ↔ ((2 · 𝑁) ∥ (𝐾 − 𝑀) ∨ (2 · 𝑁) ∥ (𝐾 − -𝑀))))
Theorem	jm2.15nn0 39607	Lemma 2.15 of [JonesMatijasevic] p. 695. Yrm is a polynomial for fixed N, so has the expected congruence property. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 𝐵) ∥ ((𝐴 Yrm 𝑁) − (𝐵 Yrm 𝑁)))
Theorem	jm2.16nn0 39608	Lemma 2.16 of [JonesMatijasevic] p. 695. This may be regarded as a special case of jm2.15nn0 39607 if Yrm is redefined as described in rmyluc 39541. (Contributed by Stefan O'Rear, 1-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → (𝐴 − 1) ∥ ((𝐴 Yrm 𝑁) − 𝑁))
20.29.33  X and Y sequences 4: Diophantine representability of Y
Theorem	jm2.27a 39609	Lemma for jm2.27 39612. Reverse direction after existential quantifiers are expanded. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐺 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐻 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)    &   ⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)    &   ⊢ (𝜑 → 𝐺 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)    &   ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    &   ⊢ (𝜑 → 𝑃 ∈ ℤ)    &   ⊢ (𝜑 → 𝐷 = (𝐴 Xrm 𝑃))    &   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝑃))    &   ⊢ (𝜑 → 𝑄 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 = (𝐴 Xrm 𝑄))    &   ⊢ (𝜑 → 𝐸 = (𝐴 Yrm 𝑄))    &   ⊢ (𝜑 → 𝑅 ∈ ℤ)    &   ⊢ (𝜑 → 𝐼 = (𝐺 Xrm 𝑅))    &   ⊢ (𝜑 → 𝐻 = (𝐺 Yrm 𝑅))    ⇒   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵))
Theorem	jm2.27b 39610	Lemma for jm2.27 39612. Expand existential quantifiers for reverse direction. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐸 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐹 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐺 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐻 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐼 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1)    &   ⊢ (𝜑 → ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1)    &   ⊢ (𝜑 → 𝐺 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1)    &   ⊢ (𝜑 → 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐺 − 𝐴))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐺 − 1))    &   ⊢ (𝜑 → 𝐹 ∥ (𝐻 − 𝐶))    &   ⊢ (𝜑 → (2 · 𝐶) ∥ (𝐻 − 𝐵))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵))
Theorem	jm2.27c 39611	Lemma for jm2.27 39612. Forward direction with substitutions. (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 = (𝐴 Yrm 𝐵))    &   ⊢ 𝐷 = (𝐴 Xrm 𝐵)    &   ⊢ 𝑄 = (𝐵 · (𝐴 Yrm 𝐵))    &   ⊢ 𝐸 = (𝐴 Yrm (2 · 𝑄))    &   ⊢ 𝐹 = (𝐴 Xrm (2 · 𝑄))    &   ⊢ 𝐺 = (𝐴 + ((𝐹↑2) · ((𝐹↑2) − 𝐴)))    &   ⊢ 𝐻 = (𝐺 Yrm 𝐵)    &   ⊢ 𝐼 = (𝐺 Xrm 𝐵)    &   ⊢ 𝐽 = ((𝐸 / (2 · (𝐶↑2))) − 1)    ⇒   ⊢ (𝜑 → (((𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0 ∧ 𝐹 ∈ ℕ0) ∧ (𝐺 ∈ ℕ0 ∧ 𝐻 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝐽 ∈ ℕ0 ∧ (((((𝐷↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1 ∧ ((𝐹↑2) − (((𝐴↑2) − 1) · (𝐸↑2))) = 1 ∧ 𝐺 ∈ (ℤ≥‘2)) ∧ (((𝐼↑2) − (((𝐺↑2) − 1) · (𝐻↑2))) = 1 ∧ 𝐸 = ((𝐽 + 1) · (2 · (𝐶↑2))) ∧ 𝐹 ∥ (𝐺 − 𝐴))) ∧ (((2 · 𝐶) ∥ (𝐺 − 1) ∧ 𝐹 ∥ (𝐻 − 𝐶)) ∧ ((2 · 𝐶) ∥ (𝐻 − 𝐵) ∧ 𝐵 ≤ 𝐶))))))
Theorem	jm2.27 39612*	Lemma 2.27 of [JonesMatijasevic] p. 697; rmY is a diophantine relation. 0 was excluded from the range of B and the lower limit of G was imposed because the source proof does not seem to work otherwise; quite possible I'm just missing something. The source proof uses both i and I; i has been changed to j to avoid collision. This theorem is basically nothing but substitution instances, all the work is done in jm2.27a 39609 and jm2.27c 39611. Once Diophantine relations have been defined, the content of the theorem is "rmY is Diophantine" (Contributed by Stefan O'Rear, 4-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 = (𝐴 Yrm 𝐵) ↔ ∃𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∃𝑓 ∈ ℕ0 ∃𝑔 ∈ ℕ0 ∃ℎ ∈ ℕ0 ∃𝑖 ∈ ℕ0 ∃𝑗 ∈ ℕ0 (((((𝑑↑2) − (((𝐴↑2) − 1) · (𝐶↑2))) = 1 ∧ ((𝑓↑2) − (((𝐴↑2) − 1) · (𝑒↑2))) = 1 ∧ 𝑔 ∈ (ℤ≥‘2)) ∧ (((𝑖↑2) − (((𝑔↑2) − 1) · (ℎ↑2))) = 1 ∧ 𝑒 = ((𝑗 + 1) · (2 · (𝐶↑2))) ∧ 𝑓 ∥ (𝑔 − 𝐴))) ∧ (((2 · 𝐶) ∥ (𝑔 − 1) ∧ 𝑓 ∥ (ℎ − 𝐶)) ∧ ((2 · 𝐶) ∥ (ℎ − 𝐵) ∧ 𝐵 ≤ 𝐶)))))
Theorem	jm2.27dlem1 39613*	Lemma for rmydioph 39618. Substitution of a tuple restriction into a projection that doesn't care. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ (1...𝐵)    ⇒   ⊢ (𝑎 = (𝑏 ↾ (1...𝐵)) → (𝑎‘𝐴) = (𝑏‘𝐴))
Theorem	jm2.27dlem2 39614	Lemma for rmydioph 39618. This theorem is used along with the next three to efficiently infer steps like 7 ∈ (1...;10). (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ (1...𝐵)    &   ⊢ 𝐶 = (𝐵 + 1)    &   ⊢ 𝐵 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ (1...𝐶)
Theorem	jm2.27dlem3 39615	Lemma for rmydioph 39618. Infer membership of the endpoint of a range. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ ℕ    ⇒   ⊢ 𝐴 ∈ (1...𝐴)
Theorem	jm2.27dlem4 39616	Lemma for rmydioph 39618. Infer ℕ-hood of large numbers. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐴 ∈ ℕ    &   ⊢ 𝐵 = (𝐴 + 1)    ⇒   ⊢ 𝐵 ∈ ℕ
Theorem	jm2.27dlem5 39617	Lemma for rmydioph 39618. Used with sselii 3966 to infer membership of midpoints of range; jm2.27dlem2 39614 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.)
⊢ 𝐵 = (𝐴 + 1)    &   ⊢ (1...𝐵) ⊆ (1...𝐶)    ⇒   ⊢ (1...𝐴) ⊆ (1...𝐶)
Theorem	rmydioph 39618	jm2.27 39612 restated in terms of Diophantine sets. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) Yrm (𝑎‘2)))} ∈ (Dioph‘3)
20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C
Theorem	rmxdiophlem 39619*	X can be expressed in terms of Y, so it is also Diophantine. (Contributed by Stefan O'Rear, 15-Oct-2014.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ℕ0) → (𝑋 = (𝐴 Xrm 𝑁) ↔ ∃𝑦 ∈ ℕ0 (𝑦 = (𝐴 Yrm 𝑁) ∧ ((𝑋↑2) − (((𝐴↑2) − 1) · (𝑦↑2))) = 1)))
Theorem	rmxdioph 39620	X is a Diophantine function. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ ((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘3) = ((𝑎‘1) Xrm (𝑎‘2)))} ∈ (Dioph‘3)
Theorem	jm3.1lem1 39621	Lemma for jm3.1 39624. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)
Theorem	jm3.1lem2 39622	Lemma for jm3.1 39624. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → (𝐾↑𝑁) < ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1))
Theorem	jm3.1lem3 39623	Lemma for jm3.1 39624. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ (𝜑 → 𝐴 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴)    ⇒   ⊢ (𝜑 → ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1) ∈ ℕ)
Theorem	jm3.1 39624	Diophantine expression for exponentiation. Lemma 3.1 of [JonesMatijasevic] p. 698. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (((𝐴 ∈ (ℤ≥‘2) ∧ 𝐾 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ) ∧ (𝐾 Yrm (𝑁 + 1)) ≤ 𝐴) → (𝐾↑𝑁) = (((𝐴 Xrm 𝑁) − ((𝐴 − 𝐾) · (𝐴 Yrm 𝑁))) mod ((((2 · 𝐴) · 𝐾) − (𝐾↑2)) − 1)))
Theorem	expdiophlem1 39625*	Lemma for expdioph 39627. Fully expanded expression for exponential. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ (𝐶 ∈ ℕ0 → (((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ) ∧ 𝐶 = (𝐴↑𝐵)) ↔ ∃𝑑 ∈ ℕ0 ∃𝑒 ∈ ℕ0 ∃𝑓 ∈ ℕ0 ((𝐴 ∈ (ℤ≥‘2) ∧ 𝐵 ∈ ℕ) ∧ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑑 = (𝐴 Yrm (𝐵 + 1))) ∧ ((𝑑 ∈ (ℤ≥‘2) ∧ 𝑒 = (𝑑 Yrm 𝐵)) ∧ ((𝑑 ∈ (ℤ≥‘2) ∧ 𝑓 = (𝑑 Xrm 𝐵)) ∧ (𝐶 < ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∧ ((((2 · 𝑑) · 𝐴) − (𝐴↑2)) − 1) ∥ ((𝑓 − ((𝑑 − 𝐴) · 𝑒)) − 𝐶))))))))
Theorem	expdiophlem2 39626	Lemma for expdioph 39627. Exponentiation on a restricted domain is Diophantine. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ (((𝑎‘1) ∈ (ℤ≥‘2) ∧ (𝑎‘2) ∈ ℕ) ∧ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2)))} ∈ (Dioph‘3)
Theorem	expdioph 39627	The exponential function is Diophantine. This result completes and encapsulates our development using Pell equation solution sequences and is sometimes regarded as Matiyasevich's theorem properly. (Contributed by Stefan O'Rear, 17-Oct-2014.)
⊢ {𝑎 ∈ (ℕ0 ↑m (1...3)) ∣ (𝑎‘3) = ((𝑎‘1)↑(𝑎‘2))} ∈ (Dioph‘3)
20.29.35  Uncategorized stuff not associated with a major project
Theorem	setindtr 39628*	Set induction for sets contained in a transitive set. If we are allowed to assume Infinity, then all sets have a transitive closure and this reduces to setind 9178; however, this version is useful without Infinity. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (∃𝑦(Tr 𝑦 ∧ 𝐵 ∈ 𝑦) → 𝐵 ∈ 𝐴))
Theorem	setindtrs 39629*	Set induction scheme without Infinity. See comments at setindtr 39628. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ⊢ (∃𝑧(Tr 𝑧 ∧ 𝐵 ∈ 𝑧) → 𝜒)
Theorem	dford3lem1 39630*	Lemma for dford3 39632. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ ((Tr 𝑁 ∧ ∀𝑦 ∈ 𝑁 Tr 𝑦) → ∀𝑏 ∈ 𝑁 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))
Theorem	dford3lem2 39631*	Lemma for dford3 39632. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)
Theorem	dford3 39632*	Ordinals are precisely the hereditarily transitive classes. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (Ord 𝑁 ↔ (Tr 𝑁 ∧ ∀𝑥 ∈ 𝑁 Tr 𝑥))
Theorem	dford4 39633*	dford3 39632 expressed in primitives to demonstrate shortness. (Contributed by Stefan O'Rear, 28-Oct-2014.)
⊢ (Ord 𝑁 ↔ ∀𝑎∀𝑏∀𝑐((𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑎) → (𝑏 ∈ 𝑁 ∧ (𝑐 ∈ 𝑏 → 𝑐 ∈ 𝑎))))
Theorem	wopprc 39634	Unrelated: Wiener pairs treat proper classes symmetrically. (Contributed by Stefan O'Rear, 19-Sep-2014.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ ¬ 1o ∈ {{{𝐴}, ∅}, {{𝐵}}})
Theorem	rpnnen3lem 39635*	Lemma for rpnnen3 39636. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ (((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) ∧ 𝑎 < 𝑏) → {𝑐 ∈ ℚ ∣ 𝑐 < 𝑎} ≠ {𝑐 ∈ ℚ ∣ 𝑐 < 𝑏})
Theorem	rpnnen3 39636	Dedekind cut injection of ℝ into 𝒫 ℚ. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ ℝ ≼ 𝒫 ℚ
20.29.36  More equivalents of the Axiom of Choice
Theorem	axac10 39637	Characterization of choice similar to dffin1-5 9812. (Contributed by Stefan O'Rear, 6-Jan-2015.)
⊢ ( ≈ “ On) = V
Theorem	harinf 39638	The Hartogs number of an infinite set is at least ω. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
⊢ ((𝑆 ∈ 𝑉 ∧ ¬ 𝑆 ∈ Fin) → ω ⊆ (har‘𝑆))
Theorem	wdom2d2 39639*	Deduction for weak dominance by a Cartesian product. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑥 = 𝑋)    ⇒   ⊢ (𝜑 → 𝐴 ≼* (𝐵 × 𝐶))
Theorem	ttac 39640	Tarski's theorem about choice: infxpidm 9986 is equivalent to ax-ac 9883. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Stefan O'Rear, 10-Jul-2015.)
⊢ (CHOICE ↔ ∀𝑐(ω ≼ 𝑐 → (𝑐 × 𝑐) ≈ 𝑐))
Theorem	pw2f1ocnv 39641*	Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8626, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))))
Theorem	pw2f1o2 39642*	Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 8626, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴)
Theorem	pw2f1o2val 39643*	Function value of the pw2f1o2 39642 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o}))
Theorem	pw2f1o2val2 39644*	Membership in a mapped set under the pw2f1o2 39642 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))    ⇒   ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o))
Theorem	soeq12d 39645	Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝜑 → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵))
Theorem	freq12d 39646	Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝜑 → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵))
Theorem	weeq12d 39647	Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝜑 → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵))
Theorem	limsuc2 39648	Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ ((Ord 𝐴 ∧ 𝐴 = ∪ 𝐴) → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴))
Theorem	wepwsolem 39649*	Transfer an ordering on characteristic functions by isomorphism to the power set. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))}    &   ⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝐹 = (𝑎 ∈ (2o ↑m 𝐴) ↦ (◡𝑎 “ {1o}))    ⇒   ⊢ (𝐴 ∈ V → 𝐹 Isom 𝑈, 𝑇 ((2o ↑m 𝐴), 𝒫 𝐴))
Theorem	wepwso 39650*	A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴 ∈ 𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴)
Theorem	dnnumch1 39651*	Define an enumeration of a set from a choice function; second part, it restricts to a bijection. EDITORIAL: overlaps dfac8a 9458. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
Theorem	dnnumch2 39652*	Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ ran 𝐹)
Theorem	dnnumch3lem 39653*	Value of the ordinal injection function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤}))
Theorem	dnnumch3 39654*	Define an injection from a set into the ordinals using a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On)
Theorem	dnwech 39655*	Define a well-ordering from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦))    &   ⊢ 𝐻 = {⟨𝑣, 𝑤⟩ ∣ ∩ (◡𝐹 “ {𝑣}) ∈ ∩ (◡𝐹 “ {𝑤})}    ⇒   ⊢ (𝜑 → 𝐻 We 𝐴)
Theorem	fnwe2val 39656*	Lemma for fnwe2 39660. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    ⇒   ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
Theorem	fnwe2lem1 39657*	Lemma for fnwe2 39660. Substitution in well-ordering hypothesis. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
Theorem	fnwe2lem2 39658*	Lemma for fnwe2 39660. An element which is in a minimal fiber and minimal within its fiber is minimal globally; thus 𝑇 is well-founded. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑎 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑎 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 ¬ 𝑐𝑇𝑏)
Theorem	fnwe2lem3 39659*	Lemma for fnwe2 39660. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑎 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑏 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑎𝑇𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑇𝑎))
Theorem	fnwe2 39660*	A well-ordering can be constructed on a partitioned set by patching together well-orderings on each partition using a well-ordering on the partitions themselves. Similar to fnwe 7828 but does not require the within-partition ordering to be globally well. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})    &   ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 We 𝐴)
Theorem	aomclem1 39661*	Lemma for dfac11 39669. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of (𝑅1‘𝐴). In what follows, 𝐴 is the index of the rank we wish to well-order, 𝑧 is the collection of well-orderings constructed so far, dom 𝑧 is the set of ordinal indices of constructed ranks i.e. the next rank to construct, and 𝑦 is a postulated multiple-choice function. Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    ⇒   ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))
Theorem	aomclem2 39662*	Lemma for dfac11 39669. Successor case 2, a choice function for subsets of (𝑅1‘dom 𝑧). (Contributed by Stefan O'Rear, 18-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎))
Theorem	aomclem3 39663*	Lemma for dfac11 39669. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧))
Theorem	aomclem4 39664*	Lemma for dfac11 39669. Limit case. Patch together well-orderings constructed so far using fnwe2 39660 to cover the limit rank. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 = ∪ dom 𝑧)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    ⇒   ⊢ (𝜑 → 𝐹 We (𝑅1‘dom 𝑧))
Theorem	aomclem5 39665*	Lemma for dfac11 39669. Combine the successor case with the limit case. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}    &   ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   ⊢ (𝜑 → dom 𝑧 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → 𝐺 We (𝑅1‘dom 𝑧))
Theorem	aomclem6 39666*	Lemma for dfac11 39669. Transfinite induction, close over 𝑧. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}    &   ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → (𝐻‘𝐴) We (𝑅1‘𝐴))
Theorem	aomclem7 39667*	Lemma for dfac11 39669. (𝑅1‘𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}    &   ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵))    &   ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))))    &   ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})}    &   ⊢ 𝐹 = {⟨𝑎, 𝑏⟩ ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))}    &   ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧)))    &   ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))
Theorem	aomclem8 39668*	Lemma for dfac11 39669. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))    ⇒   ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))
Theorem	dfac11 39669*	The right-hand side of this theorem (compare with ac4 9899), sometimes known as the "axiom of multiple choice", is a choice equivalent. Curiously, this statement cannot be proved without ax-reg 9058, despite not mentioning the cumulative hierarchy in any way as most consequences of regularity do. This is definition (MC) of [Schechter] p. 141. EDITORIAL: the proof is not original with me of course but I lost my reference sometime after writing it. A multiple choice function allows any total order to be extended to a choice function, which in turn defines a well-ordering. Since a well-ordering on a set defines a simple ordering of the power set, this allows the trivial well-ordering of the empty set to be transfinitely bootstrapped up the cumulative hierarchy to any desired level. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Stefan O'Rear, 1-Jun-2015.)
⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ ((𝒫 𝑧 ∩ Fin) ∖ {∅})))
Theorem	kelac1 39670*	Kelley's choice, basic form: if a collection of sets can be cast as closed sets in the factors of a topology, and there is a definable element in each topology (which need not be in the closed set - if it were this would be trivial), then compactness (via finite intersection) guarantees that the final product is nonempty. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐶 ∈ (Clsd‘𝐽))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐵:𝑆–1-1-onto→𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑈 ∈ ∪ 𝐽)    &   ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ 𝐽)) ∈ Comp)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
Theorem	kelac2lem 39671	Lemma for kelac2 39672 and dfac21 39673: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
Theorem	kelac2 39672*	Kelley's choice, most common form: compactness of a product of knob topologies recovers choice. (Contributed by Stefan O'Rear, 22-Feb-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ≠ ∅)    &   ⊢ (𝜑 → (∏t‘(𝑥 ∈ 𝐼 ↦ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}))) ∈ Comp)    ⇒   ⊢ (𝜑 → X𝑥 ∈ 𝐼 𝑆 ≠ ∅)
Theorem	dfac21 39673	Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
20.29.37  Finitely generated left modules
Syntax	clfig 39674	Extend class notation with the class of finitely generated left modules.
class LFinGen
Definition	df-lfig 39675	Define the class of finitely generated left modules. Finite generation of subspaces can be intepreted using ↾s. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫 (Base‘𝑤) ∩ Fin))}
Theorem	islmodfg 39676*	Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝐵(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝐵)))
Theorem	islssfg 39677*	Property of a finitely generated left (sub)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ 𝒫 𝑈(𝑏 ∈ Fin ∧ (𝑁‘𝑏) = 𝑈)))
Theorem	islssfg2 39678*	Property of a finitely generated left (sub)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑋 ∈ LFinGen ↔ ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)(𝑁‘𝑏) = 𝑈))
Theorem	islssfgi 39679	Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s (𝑁‘𝐵))    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ⊆ 𝑉 ∧ 𝐵 ∈ Fin) → 𝑋 ∈ LFinGen)
Theorem	fglmod 39680	Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod)
Theorem	lsmfgcl 39681	The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐷 = (𝑊 ↾s 𝐴)    &   ⊢ 𝐸 = (𝑊 ↾s 𝐵)    &   ⊢ 𝐹 = (𝑊 ↾s (𝐴 ⊕ 𝐵))    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ LFinGen)    &   ⊢ (𝜑 → 𝐸 ∈ LFinGen)    ⇒   ⊢ (𝜑 → 𝐹 ∈ LFinGen)
20.29.38  Noetherian left modules I
Syntax	clnm 39682	Extend class notation with the class of Noetherian left modules.
class LNoeM
Definition	df-lnm 39683*	A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen}
Theorem	islnm 39684*	Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ 𝑆 = (LSubSp‘𝑀)    ⇒   ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen))
Theorem	islnm2 39685*	Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑁 = (LSpan‘𝑀)    ⇒   ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔)))
Theorem	lnmlmod 39686	A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LMod)
Theorem	lnmlssfg 39687	A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (𝑀 ↾s 𝑈)    ⇒   ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)
Theorem	lnmlsslnm 39688	All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (𝑀 ↾s 𝑈)    ⇒   ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LNoeM)
Theorem	lnmfg 39689	A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
⊢ (𝑀 ∈ LNoeM → 𝑀 ∈ LFinGen)
Theorem	kercvrlsm 39690	The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑈 = (LSubSp‘𝑆)    &   ⊢ ⊕ = (LSSum‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐹 “ 𝐷) = ran 𝐹)    ⇒   ⊢ (𝜑 → (𝐾 ⊕ 𝐷) = 𝐵)
Theorem	lmhmfgima 39691	A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑇 ↾s (𝐹 “ 𝐴))    &   ⊢ 𝑋 = (𝑆 ↾s 𝐴)    &   ⊢ 𝑈 = (LSubSp‘𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ LFinGen)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    ⇒   ⊢ (𝜑 → 𝑌 ∈ LFinGen)
Theorem	lnmepi 39692	Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐵 = (Base‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑆 ∈ LNoeM ∧ ran 𝐹 = 𝐵) → 𝑇 ∈ LNoeM)
Theorem	lmhmfgsplit 39693	If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑈 = (𝑆 ↾s 𝐾)    &   ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)
Theorem	lmhmlnmsplit 39694	If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑈 = (𝑆 ↾s 𝐾)    &   ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
Theorem	lnmlmic 39695	Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑚 𝑆 → (𝑅 ∈ LNoeM ↔ 𝑆 ∈ LNoeM))
20.29.39  Addenda for structure powers
Theorem	pwssplit4 39696*	Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐸 = (𝑅 ↑s (𝐴 ∪ 𝐵))    &   ⊢ 𝐺 = (Base‘𝐸)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐾 = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })}    &   ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵))    &   ⊢ 𝐶 = (𝑅 ↑s 𝐴)    &   ⊢ 𝐷 = (𝑅 ↑s 𝐵)    &   ⊢ 𝐿 = (𝐸 ↾s 𝐾)    ⇒   ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Theorem	filnm 39697	Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM)
Theorem	pwslnmlem0 39698	Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s ∅)    ⇒   ⊢ (𝑊 ∈ LMod → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem1 39699*	First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝑌 = (𝑊 ↑s {𝑖})    ⇒   ⊢ (𝑊 ∈ LNoeM → 𝑌 ∈ LNoeM)
Theorem	pwslnmlem2 39700	A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑋 = (𝑊 ↑s 𝐴)    &   ⊢ 𝑌 = (𝑊 ↑s 𝐵)    &   ⊢ 𝑍 = (𝑊 ↑s (𝐴 ∪ 𝐵))    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ LNoeM)    &   ⊢ (𝜑 → 𝑌 ∈ LNoeM)    ⇒   ⊢ (𝜑 → 𝑍 ∈ LNoeM)