P. 431 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	uvcn0 43001	A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ 0 )
Theorem	psrmnd 43002	The ring of power series is a monoid. (Contributed by SN, 25-Apr-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝑆 ∈ Mnd)
Theorem	psrbagres 43003*	Restrict a bag of variables in 𝐼 to a bag of variables in 𝐽 ⊆ 𝐼. (Contributed by SN, 10-Mar-2025.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐽) ∈ 𝐸)
Theorem	mplcrngd 43004	The polynomial ring is a commutative ring. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝑃 ∈ CRing)
Theorem	mplsubrgcl 43005	An element of a polynomial algebra over a subring is an element of the polynomial algebra. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐶 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐶)
Theorem	mhmcopsr 43006	The composition of a monoid homomorphism and a power series is a power series. (Contributed by SN, 18-May-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)
Theorem	mhmcoaddpsr 43007	Show that the ring homomorphism in rhmpsr 43009 preserves addition. (Contributed by SN, 18-May-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ + = (+g‘𝑃)    &   ⊢ ✚ = (+g‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)))
Theorem	rhmcomulpsr 43008	Show that the ring homomorphism in rhmpsr 43009 preserves multiplication. (Contributed by SN, 18-May-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ∙ = (.r‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 · 𝐺)) = ((𝐻 ∘ 𝐹) ∙ (𝐻 ∘ 𝐺)))
Theorem	rhmpsr 43009*	Provide a ring homomorphism between two power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
Theorem	rhmpsr1 43010*	Provide a ring homomorphism between two univariate power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (PwSer1‘𝑅)    &   ⊢ 𝑄 = (PwSer1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
Theorem	mplmapghm 43011*	The function 𝐻 mapping polynomials 𝑝 to their coefficient given a bag of variables 𝐹 is a group homomorphism. (Contributed by SN, 15-Mar-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (𝑝 ∈ 𝐵 ↦ (𝑝‘𝐹))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝑃 GrpHom 𝑅))
Theorem	evl0 43012	The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑂 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → (𝑄‘ 0 ) = ((𝐵 ↑m 𝐼) × {𝑂}))
Theorem	evlscl 43013	A polynomial over the ring 𝑅 evaluates to an element in 𝑅. (Contributed by SN, 12-Mar-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝑆)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) ∈ 𝐾)
Theorem	evlsscaval 43014	Polynomial evaluation builder for a scalar. Compare evl1scad 22310. Note that scalar multiplication by 𝑋 is the same as vector multiplication by (𝐴‘𝑋) by asclmul1 21876. (Contributed by SN, 27-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑅)    &   ⊢ (𝜑 → 𝐿 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝐴‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋))
Theorem	evlsvarval 43015	Polynomial evaluation builder for a variable. (Contributed by SN, 27-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑉 = (𝐼 mVar 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉‘𝑋))‘𝐴) = (𝐴‘𝑋)))
Theorem	evlsbagval 43016*	Polynomial evaluation builder for a bag of variables. EDITORIAL: This theorem should stay in my mathbox until there's another use, since 0 and 1 using 𝑈 instead of 𝑆 may not be convenient. (Contributed by SN, 29-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑊 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 1 = (1r‘𝑈)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐹 = (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝐵, 1 , 0 ))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ ((𝑄‘𝐹)‘𝐴) = (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))))
Theorem	evlsexpval 43017	Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))    &   ⊢ ∙ = (.g‘(mulGrp‘𝑃))    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))
Theorem	evlsaddval 43018	Polynomial evaluation builder for addition. (Contributed by SN, 27-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))    &   ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊))    &   ⊢ ✚ = (+g‘𝑃)    &   ⊢ + = (+g‘𝑆)    ⇒   ⊢ (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (𝑉 + 𝑊)))
Theorem	evlsmulval 43019	Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))    &   ⊢ (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊))    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑆)    ⇒   ⊢ (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)))
Theorem	evlsmaprhm 43020*	The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝑋 is a ring homomorphism. Compare evls1maprhm 22351. (Contributed by SN, 12-Mar-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝑆)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑅 ↾s 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ ((𝑄‘𝑝)‘𝐴))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))
Theorem	evlsevl 43021	Evaluation in a subring is the same as evaluation in the ring itself. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑂 = (𝐼 eval 𝑆)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑄‘𝐹) = (𝑂‘𝐹))
Theorem	evlvvval 43022*	Give a formula for the evaluation of a polynomial given assignments from variables to values. (Contributed by SN, 5-Mar-2025.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑅 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
Theorem	evlvvvallem 43023*	Lemma for theorems using evlvvval 43022. Version of evlsvvvallem2 22080 using df-evl 22063. (Contributed by SN, 11-Mar-2025.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0g‘𝑅))
Theorem	selvcllem1 43024	𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼 ∖ 𝐽) and we have 𝐽 ∈ 𝑊 instead of 𝐽 ⊆ 𝐼. TODO-SN: In practice, this "simplification" makes the lemmas harder to use. (Contributed by SN, 15-Dec-2023.)
⊢ 𝑈 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝑇 ∈ AssAlg)
Theorem	selvcllem2 43025	𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
⊢ 𝑈 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝐷 ∈ (𝑅 RingHom 𝑇))
Theorem	selvcllem3 43026	The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
⊢ 𝑈 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))
Theorem	selvcllemh 43027	Apply the third argument (selvcllem3 43026) to show that 𝑄 is a (ring) homomorphism. (Contributed by SN, 5-Nov-2023.)
⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   ⊢ 𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑆)    &   ⊢ 𝑆 = (𝑇 ↾s ran 𝐷)    &   ⊢ 𝑋 = (𝑇 ↑s (𝐵 ↑m 𝐼))    &   ⊢ 𝐵 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom 𝑋))
Theorem	selvcllem4 43028	The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼 ∖ 𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   ⊢ 𝑆 = (𝑇 ↾s ran 𝐷)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑆)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷 ∘ 𝐹) ∈ 𝑋)
Theorem	selvcllem5 43029*	The fifth argument passed to evalSub is in the domain (a function 𝐼⟶𝐸). (Contributed by SN, 22-Feb-2024.)
⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ 𝐸 = (Base‘𝑇)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐸 ↑m 𝐼))
Theorem	selvcl 43030	Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐸 = (Base‘𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸)
Theorem	selvval2 43031*	Value of the "variable selection" function. Convert selvval 22111 into a simpler form by using evlsevl 43021. (Contributed by SN, 9-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐶 = (algSc‘𝑇)    &   ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = (((𝐼 eval 𝑇)‘(𝐷 ∘ 𝐹))‘(𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼 ∖ 𝐽) mVar 𝑅)‘𝑥))))))
Theorem	selvvvval 43032*	Recover the original polynomial from a selectVars application. (Contributed by SN, 15-Mar-2025.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹)‘(𝑌 ↾ 𝐽))‘(𝑌 ↾ (𝐼 ∖ 𝐽))) = (𝐹‘𝑌))
Theorem	evlselvlem 43033*	Lemma for evlselv 43034. Used to re-index to and from bags of variables in 𝐼 and bags of variables in the subsets 𝐽 and 𝐼 ∖ 𝐽. (Contributed by SN, 10-Mar-2025.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}    &   ⊢ 𝐶 = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (𝑐 ∈ 𝐶, 𝑒 ∈ 𝐸 ↦ (𝑐 ∪ 𝑒))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)
Theorem	evlselv 43034	Evaluating a selection of variable assignments, then evaluating the rest of the variables, is the same as evaluating with all assignments. (Contributed by SN, 10-Mar-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐿 = (algSc‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))
Theorem	selvadd 43035	The "variable selection" function is additive. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ + = (+g‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ ✚ = (+g‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐹 + 𝐺)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ✚ (((𝐼 selectVars 𝑅)‘𝐽)‘𝐺)))
Theorem	selvmul 43036	The "variable selection" function is multiplicative. (Contributed by SN, 18-Feb-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ ∙ = (.r‘𝑇)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐹 · 𝐺)) = ((((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∙ (((𝐼 selectVars 𝑅)‘𝐽)‘𝐺)))
Theorem	fsuppind 43037*	Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 21746 and frlmplusgval 21754). This theorem is structurally general for polynomial proof usage (see mplelbas 21979 and mpladd 21997). Note that hypothesis 0 is redundant when 𝐼 is nonempty. (Contributed by SN, 18-May-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘f + 𝑦) ∈ 𝐻)    ⇒   ⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)
Theorem	fsuppssindlem1 43038*	Lemma for fsuppssind 43040. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))
Theorem	fsuppssindlem2 43039*	Lemma for fsuppssind 43040. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
Theorem	fsuppssind 43040*	Induction on functions 𝐹:𝐴⟶𝐵 with finite support (see fsuppind 43037) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐼)    &   ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘f + 𝑦) ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑋 finSupp 0 )    &   ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐻)
Theorem	mhpind 43041*	The homogeneous polynomials of degree 𝑁 are generated by the terms of degree 𝑁 and addition. (Contributed by SN, 28-Jul-2024.)
⊢ 𝐻 = (𝐼 mHomP 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ + = (+g‘𝑃)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐺)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐺)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐻‘𝑁) ∩ 𝐺) ∧ 𝑦 ∈ ((𝐻‘𝑁) ∩ 𝐺))) → (𝑥 + 𝑦) ∈ 𝐺)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐺)
Theorem	evlsmhpvvval 43042*	Give a formula for the evaluation of a homogeneous polynomial given assignments from variables to values. The difference between this and evlsvvval 22081 is that 𝑏 ∈ 𝐷 is restricted to 𝑏 ∈ 𝐺, that is, we can evaluate an 𝑁-th degree homogeneous polynomial over just the terms where the sum of all variable degrees is 𝑁. (Contributed by SN, 5-Mar-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
Theorem	mhphflem 43043*	Lemma for mhphf 43044. Add several multiples of 𝐿 together, in a case where the total amount of multiplies is 𝑁. (Contributed by SN, 30-Jul-2024.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐻) → (𝐺 Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) · 𝐿))) = (𝑁 · 𝐿))
Theorem	mhphf 43044	A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄‘𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.) (Proof shortened by SN, 8-Mar-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
Theorem	mhphf2 43045	A homogeneous polynomial defines a homogeneous function; this is mhphf 43044 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 21756 but without the finite support restriction (frlmpws 21740, frlmbas 21745) on the assignments 𝐴 from variables to values. TODO?: Polynomials (df-mpl 21901) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ ∙ = ( ·𝑠 ‘((ringLMod‘𝑆) ↑s 𝐼))    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
Theorem	mhphf3 43046	A homogeneous polynomial defines a homogeneous function; this is mhphf2 43045 with the finite support restriction (frlmpws 21740, frlmbas 21745) on the assignments 𝐴 from variables to values. See comment of mhphf2 43045. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐹 = (𝑆 freeLMod 𝐼)    &   ⊢ 𝑀 = (Base‘𝐹)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
Theorem	mhphf4 43047	A homogeneous polynomial defines a homogeneous function; this is mhphf3 43046 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = (𝐼 eval 𝑆)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑆)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐹 = (𝑆 freeLMod 𝐼)    &   ⊢ 𝑀 = (Base‘𝐹)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
21.30.7  Projective spaces Looking at a corner in 3D space, one can see three right angles. It is impossible to draw three lines in 2D space such that any two of these lines are perpendicular, but a good enough representation is made by casting lines from the 2D surface. Points along the same cast line are collapsed into one point on the 2D surface. In many cases, the 2D surface is smaller than whatever needs to be represented. If the lines cast were perpendicular to the 2D surface, then only areas as small as the 2D surface could be represented. To fix this, the lines need to get further apart as they go farther from the 2D surface. On the other side of the 2D surface the lines will get closer together and intersect at a point (because it's defined that way). From this perspective, two parallel lines in 3D space will be represented by two lines that seem to intersect at a point "at infinity". Considering all maximal classes of parallel lines on a 2D plane in 3D space, these classes will all appear to intersect at different points at infinity, forming a line at infinity. Therefore the real projective plane can be thought of as the real affine plane together with the line at infinity. The projective plane takes care of some exceptions that may be found in the affine plane. For example, consider the curve that is the zeroes of 𝑦 = 𝑥↑2. Any line connecting the point (0, 1) to the x-axis intersects with the curve twice, except for the vertical line between (0, 1) and (0, 0). In the projective plane, the curve becomes an ellipse and there is no exception. While it may not seem like it, points at infinity and points corresponding to the affine plane are the same type of point. Consider a line going through the origin in 3D (affine) space. Either it intersects the plane 𝑧 = 1 once, or it is entirely within the plane 𝑧 = 0. If it is entirely within the plane 𝑧 = 0, then it corresponds to the point at infinity intersecting all lines on the plane 𝑧 = 1 with the same slope. Else it corresponds to the point in the 2D plane 𝑧 = 1 that it intersects. So there is a bijection between 3D lines through the origin and points on the real projective plane. The concept of projective spaces generalizes the projective plane to any dimension.
Syntax	cprjsp 43048	Extend class notation with the projective space function.
class ℙ𝕣𝕠𝕛
Definition	df-prjsp 43049*	Define the projective space function. In the bijection between 3D lines through the origin and points in the projective plane (see section comment), this is equivalent to making any two 3D points (excluding the origin) equivalent iff one is a multiple of another. This definition does not quite give all the properties needed, since the scalars of a left vector space can be "less dense" than the vectors (for example, making equivalent rational multiples of real numbers). Compare df-lsatoms 39436. (Contributed by BJ and SN, 29-Apr-2023.)
⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
Theorem	prjspval 43050*	Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))
Theorem	prjsprel 43051*	Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    ⇒   ⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌)))
Theorem	prjspertr 43052*	The relation in ℙ𝕣𝕠𝕛 is transitive. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LMod ∧ (𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍)) → 𝑋 ∼ 𝑍)
Theorem	prjsperref 43053*	The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ 𝑋 ∼ 𝑋))
Theorem	prjspersym 43054*	The relation in ℙ𝕣𝕠𝕛 is symmetric. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∼ 𝑌) → 𝑌 ∼ 𝑋)
Theorem	prjsper 43055*	The relation used to define ℙ𝕣𝕠𝕛 is an equivalence relation. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → ∼ Er 𝐵)
Theorem	prjspreln0 43056*	Two nonzero vectors are equivalent by a nonzero scalar. (Contributed by Steven Nguyen, 31-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑚 · 𝑌))))
Theorem	prjspvs 43057*	A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∼ 𝑋)
Theorem	prjsprellsp 43058*	Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∼ 𝑌 ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
Theorem	prjspeclsp 43059*	The vectors equivalent to a vector 𝑋 are the nonzero vectors in the span of 𝑋. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵) → [𝑋] ∼ = ((𝑁‘{𝑋}) ∖ {(0g‘𝑉)}))
Theorem	prjspval2 43060*	Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ 0 = (0g‘𝑉)    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ { 0 })    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = ∪ 𝑧 ∈ 𝐵 {((𝑁‘{𝑧}) ∖ { 0 })})
Syntax	cprjspn 43061	Extend class notation with the n-dimensional projective space function.
class ℙ𝕣𝕠𝕛n
Definition	df-prjspn 43062*	Define the n-dimensional projective space function. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Compare df-ehl 25363. This space is considered n-dimensional because the vector space (𝑘 freeLMod (0...𝑛)) is (n+1)-dimensional and the ℙ𝕣𝕠𝕛 function returns equivalence classes with respect to a linear (1-dimensional) relation. (Contributed by BJ and Steven Nguyen, 29-Apr-2023.)
⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
Theorem	prjspnval 43063	Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
Theorem	prjspnerlem 43064*	A lemma showing that the equivalence relation used in prjspnval2 43065 and the equivalence relation used in prjspval 43050 are equal, but only with the antecedent 𝐾 ∈ DivRing. (Contributed by SN, 15-Jul-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
Theorem	prjspnval2 43065*	Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ ))
Theorem	prjspner 43066*	The relation used to define ℙ𝕣𝕠𝕛 (and indirectly ℙ𝕣𝕠𝕛n through df-prjspn 43062) is an equivalence relation. This is a lemma that converts the equivalence relation used in results like prjspertr 43052 and prjspersym 43054 (see prjspnerlem 43064). Several theorems are covered in one thanks to the theorems around df-er 8636. (Contributed by SN, 14-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    ⇒   ⊢ (𝜑 → ∼ Er 𝐵)
Theorem	prjspnvs 43067*	A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 43057 (see prjspnerlem 43064). (Contributed by SN, 8-Aug-2024.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋)
Theorem	prjspnssbas 43068	A projective point spans a subset of the (nonzero) affine points. (Contributed by SN, 17-Jan-2025.)
⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑃 ⊆ 𝒫 𝐵)
Theorem	prjspnn0 43069	A projective point is nonempty. (Contributed by SN, 17-Jan-2025.)
⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	0prjspnlem 43070	Lemma for 0prjspn 43075. The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 1 = ((𝐾 unitVec (0...0))‘0)    ⇒   ⊢ (𝐾 ∈ DivRing → 1 ∈ 𝐵)
Theorem	prjspnfv01 43071*	Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the value of the zeroth coordinate). (Contributed by SN, 13-Aug-2023.)
⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 1 = (1r‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑋)‘0) = if((𝑋‘0) = 0 , 0 , 1 ))
Theorem	prjspner01 43072*	Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the equivalence). (Contributed by SN, 13-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))
Theorem	prjspner1 43073*	Two vectors whose zeroth coordinate is nonzero are equivalent if and only if they have the same representative in the (n-1)-dimensional affine subspace { x0 = 1 } . For example, vectors in 3D space whose 𝑥 coordinate is nonzero are equivalent iff they intersect at the plane 𝑥 = 1 at the same point (also see section header). (Contributed by SN, 13-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋‘0) ≠ 0 )    &   ⊢ (𝜑 → (𝑌‘0) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
Theorem	0prjspnrel 43074*	In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 1 = ((𝐾 unitVec (0...0))‘0)    ⇒   ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∼ 1 )
Theorem	0prjspn 43075	A zero-dimensional projective space has only 1 point. (Contributed by Steven Nguyen, 9-Jun-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    ⇒   ⊢ (𝐾 ∈ DivRing → (0ℙ𝕣𝕠𝕛n𝐾) = {𝐵})
Syntax	cprjcrv 43076	Extend class notation with the projective curve function.
class ℙ𝕣𝕠𝕛Crv
Definition	df-prjcrv 43077*	Define the projective curve function. This takes a homogeneous polynomial and outputs the homogeneous coordinates where the polynomial evaluates to zero (the "zero set"). (In other words, scalar multiples are collapsed into the same projective point. See mhphf4 43047 and prjspvs 43057). (Contributed by SN, 23-Nov-2024.)
⊢ ℙ𝕣𝕠𝕛Crv = (𝑛 ∈ ℕ0, 𝑘 ∈ Field ↦ (𝑓 ∈ ∪ ran ((0...𝑛) mHomP 𝑘) ↦ {𝑝 ∈ (𝑛ℙ𝕣𝕠𝕛n𝑘) ∣ ((((0...𝑛) eval 𝑘)‘𝑓) “ 𝑝) = {(0g‘𝑘)}}))
Theorem	prjcrvfval 43078*	Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)    &   ⊢ 𝐸 = ((0...𝑁) eval 𝐾)    &   ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    ⇒   ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛Crv𝐾) = (𝑓 ∈ ∪ ran 𝐻 ↦ {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝑓) “ 𝑝) = { 0 }}))
Theorem	prjcrvval 43079*	Value of the projective curve function. (Contributed by SN, 23-Nov-2024.)
⊢ 𝐻 = ((0...𝑁) mHomP 𝐾)    &   ⊢ 𝐸 = ((0...𝑁) eval 𝐾)    &   ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    &   ⊢ (𝜑 → 𝐹 ∈ ∪ ran 𝐻)    ⇒   ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘𝐹) = {𝑝 ∈ 𝑃 ∣ ((𝐸‘𝐹) “ 𝑝) = { 0 }})
Theorem	prjcrv0 43080	The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑌 = ((0...𝑁) mPoly 𝐾)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ Field)    ⇒   ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃)
21.30.8  Basic reductions for Fermat's Last Theorem
Theorem	dffltz 43081*	Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes (𝑎↑𝑛) + (𝑏↑𝑛) = (𝑐↑𝑛), and adds the negative of any negative term to both sides, thus creating the corresponding equation with only positive integers. There are six combinations of negativity, so the proof is particularly long. (Contributed by Steven Nguyen, 27-Feb-2023.)
⊢ (∀𝑛 ∈ (ℤ≥‘3)∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ ∀𝑧 ∈ ℕ ((𝑥↑𝑛) + (𝑦↑𝑛)) ≠ (𝑧↑𝑛) ↔ ∀𝑛 ∈ (ℤ≥‘3)∀𝑎 ∈ (ℤ ∖ {0})∀𝑏 ∈ (ℤ ∖ {0})∀𝑐 ∈ (ℤ ∖ {0})((𝑎↑𝑛) + (𝑏↑𝑛)) ≠ (𝑐↑𝑛))
Theorem	fltmul 43082	A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (There does not seem to be a standard term for Fermat or Pythagorean triples extended to any 𝑁 ∈ ℕ0, so the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁))
Theorem	fltdiv 43083	A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝑆 ≠ 0)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁))
Theorem	flt0 43084	A counterexample for FLT does not exist for 𝑁 = 0. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ)
Theorem	fltdvdsabdvdsc 43085	Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 43086. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶)
Theorem	fltabcoprmex 43086	A counterexample to FLT implies a counterexample to FLT with 𝐴, 𝐵 (assigned to 𝐴 / (𝐴 gcd 𝐵) and 𝐵 / (𝐴 gcd 𝐵)) coprime (by divgcdcoprm0 16625). (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → (((𝐴 / (𝐴 gcd 𝐵))↑𝑁) + ((𝐵 / (𝐴 gcd 𝐵))↑𝑁)) = ((𝐶 / (𝐴 gcd 𝐵))↑𝑁))
Theorem	fltaccoprm 43087	A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐴, 𝐶 coprime. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)
Theorem	fltbccoprm 43088	A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐵, 𝐶 coprime. Proven from fltaccoprm 43087 using commutativity of addition. (Contributed by SN, 20-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 gcd 𝐶) = 1)
Theorem	fltabcoprm 43089	A counterexample to FLT with 𝐴, 𝐶 coprime also has 𝐴, 𝐵 coprime. Converse of fltaccoprm 43087. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)
Theorem	infdesc 43090*	Infinite descent. The hypotheses say that 𝑆 is lower bounded, and that if 𝜓 holds for an integer in 𝑆, it holds for a smaller integer in 𝑆. By infinite descent, eventually we cannot go any smaller, therefore 𝜓 holds for no integer in 𝑆. (Contributed by SN, 20-Aug-2024.)
⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑆 ⊆ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝜒)) → ∃𝑧 ∈ 𝑆 (𝜃 ∧ 𝑧 < 𝑥))    ⇒   ⊢ (𝜑 → {𝑦 ∈ 𝑆 ∣ 𝜓} = ∅)
Theorem	fltne 43091	If a counterexample to FLT exists, its addends are not equal. (Contributed by SN, 1-Jun-2023.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	flt4lem 43092	Raising a number to the fourth power is equivalent to squaring it twice. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2))
Theorem	flt4lem1 43093	Satisfy the antecedent used in several pythagtrip 16796 lemmas, with 𝐴, 𝐶 coprime rather than 𝐴, 𝐵. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)))
Theorem	flt4lem2 43094	If 𝐴 is even, 𝐵 is odd. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ¬ 2 ∥ 𝐵)
Theorem	flt4lem3 43095	Equivalent to pythagtriplem4 16781. Show that 𝐶 + 𝐴 and 𝐶 − 𝐴 are coprime. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1)
Theorem	flt4lem4 43096	If the product of two coprime factors is a perfect square, the factors are perfect squares. (Contributed by SN, 22-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐶↑2))    ⇒   ⊢ (𝜑 → (𝐴 = ((𝐴 gcd 𝐶)↑2) ∧ 𝐵 = ((𝐵 gcd 𝐶)↑2)))
Theorem	flt4lem5 43097	In the context of the lemmas of pythagtrip 16796, 𝑀 and 𝑁 are coprime. (Contributed by SN, 23-Aug-2024.)
⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)    &   ⊢ 𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)    ⇒   ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)
Theorem	flt4lem5elem 43098	Version of fltaccoprm 43087 and fltbccoprm 43088 where 𝑀 is not squared. This can be proved in general for any polynomial in three variables: using prmdvdsncoprmbd 16688, dvds2addd 16252, and prmdvdsexp 16676, we can show that if two variables are coprime, the third is also coprime to the two. (Contributed by SN, 24-Aug-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))    &   ⊢ (𝜑 → (𝑅 gcd 𝑆) = 1)    ⇒   ⊢ (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))
Theorem	flt4lem5a 43099	Part 1 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   ⊢ 𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2))
Theorem	flt4lem5b 43100	Part 2 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
⊢ 𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   ⊢ 𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   ⊢ 𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ 𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   ⊢ (𝜑 → 𝐴 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵 ∈ ℕ)    &   ⊢ (𝜑 → 𝐶 ∈ ℕ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐴)    &   ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1)    &   ⊢ (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   ⊢ (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2))