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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremriccrng1 41401 Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.)
((𝑅 β‰ƒπ‘Ÿ 𝑆 ∧ 𝑅 ∈ CRing) β†’ 𝑆 ∈ CRing)
 
Theoremriccrng 41402 A ring is commutative if and only if an isomorphic ring is commutative. (Contributed by SN, 10-Jan-2025.)
(𝑅 β‰ƒπ‘Ÿ 𝑆 β†’ (𝑅 ∈ CRing ↔ 𝑆 ∈ CRing))
 
Theoremdrnginvrn0d 41403 A multiplicative inverse in a division ring is nonzero. (recne0d 11989 analog). (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 β‰  0 )    β‡’   (πœ‘ β†’ (πΌβ€˜π‘‹) β‰  0 )
 
Theoremdrngmulcanad 41404 Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11854 analog). (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 β‰  0 )    &   (πœ‘ β†’ (𝑍 Β· 𝑋) = (𝑍 Β· π‘Œ))    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremdrngmulcan2ad 41405 Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11855 analog). (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 β‰  0 )    &   (πœ‘ β†’ (𝑋 Β· 𝑍) = (π‘Œ Β· 𝑍))    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremdrnginvmuld 41406 Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 β‰  0 )    &   (πœ‘ β†’ π‘Œ β‰  0 )    β‡’   (πœ‘ β†’ (πΌβ€˜(𝑋 Β· π‘Œ)) = ((πΌβ€˜π‘Œ) Β· (πΌβ€˜π‘‹)))
 
Theoremricdrng1 41407 A ring isomorphism maps a division ring to a division ring. (Contributed by SN, 18-Feb-2025.)
((𝑅 β‰ƒπ‘Ÿ 𝑆 ∧ 𝑅 ∈ DivRing) β†’ 𝑆 ∈ DivRing)
 
Theoremricdrng 41408 A ring is a division ring if and only if an isomorphic ring is a division ring. (Contributed by SN, 18-Feb-2025.)
(𝑅 β‰ƒπ‘Ÿ 𝑆 β†’ (𝑅 ∈ DivRing ↔ 𝑆 ∈ DivRing))
 
Theoremricfld 41409 A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025.)
(𝑅 β‰ƒπ‘Ÿ 𝑆 β†’ (𝑅 ∈ Field ↔ 𝑆 ∈ Field))
 
Theoremlvecgrp 41410 A vector space is a group. (Contributed by SN, 28-May-2023.)
(π‘Š ∈ LVec β†’ π‘Š ∈ Grp)
 
Theoremlvecring 41411 The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
𝐹 = (Scalarβ€˜π‘Š)    β‡’   (π‘Š ∈ LVec β†’ 𝐹 ∈ Ring)
 
Theoremfrlm0vald 41412 All coordinates of the zero vector are zero. (Contributed by SN, 14-Aug-2024.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝐽 ∈ 𝐼)    β‡’   (πœ‘ β†’ ((0gβ€˜πΉ)β€˜π½) = 0 )
 
Theoremfrlmsnic 41413* Given a free module with a singleton as the index set, that is, a free module of one-dimensional vectors, the function that maps each vector to its coordinate is a module isomorphism from that module to its ring of scalars seen as a module. (Contributed by Steven Nguyen, 18-Aug-2023.)
π‘Š = (𝐾 freeLMod {𝐼})    &   πΉ = (π‘₯ ∈ (Baseβ€˜π‘Š) ↦ (π‘₯β€˜πΌ))    β‡’   ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) β†’ 𝐹 ∈ (π‘Š LMIso (ringLModβ€˜πΎ)))
 
Theoremuvccl 41414 A unit vector is a vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
π‘ˆ = (𝑅 unitVec 𝐼)    &   π‘Œ = (𝑅 freeLMod 𝐼)    &   π΅ = (Baseβ€˜π‘Œ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘Š ∧ 𝐽 ∈ 𝐼) β†’ (π‘ˆβ€˜π½) ∈ 𝐡)
 
Theoremuvcn0 41415 A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.)
π‘ˆ = (𝑅 unitVec 𝐼)    &   π‘Œ = (𝑅 freeLMod 𝐼)    &   π΅ = (Baseβ€˜π‘Œ)    &    0 = (0gβ€˜π‘Œ)    β‡’   ((𝑅 ∈ NzRing ∧ 𝐼 ∈ π‘Š ∧ 𝐽 ∈ 𝐼) β†’ (π‘ˆβ€˜π½) β‰  0 )
 
Theorempwselbasr 41416 The reverse direction of pwselbasb 17439: a function between the index and base set of a structure is an element of the structure power. (Contributed by SN, 29-Jul-2024.)
π‘Œ = (𝑅 ↑s 𝐼)    &   π΅ = (Baseβ€˜π‘…)    &   π‘‰ = (Baseβ€˜π‘Œ)    &   (πœ‘ β†’ 𝑅 ∈ π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    &   (πœ‘ β†’ 𝑋:𝐼⟢𝐡)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝑉)
 
Theorempwsgprod 41417* Finite products in a power structure are taken componentwise. Compare pwsgsum 19892. (Contributed by SN, 30-Jul-2024.)
π‘Œ = (𝑅 ↑s 𝐼)    &   π΅ = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘Œ)    &   π‘€ = (mulGrpβ€˜π‘Œ)    &   π‘‡ = (mulGrpβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ (𝑦 ∈ 𝐽 ↦ (π‘₯ ∈ 𝐼 ↦ π‘ˆ)) finSupp 1 )    β‡’   (πœ‘ β†’ (𝑀 Ξ£g (𝑦 ∈ 𝐽 ↦ (π‘₯ ∈ 𝐼 ↦ π‘ˆ))) = (π‘₯ ∈ 𝐼 ↦ (𝑇 Ξ£g (𝑦 ∈ 𝐽 ↦ π‘ˆ))))
 
Theorempsrbagres 41418* Restrict a bag of variables in 𝐼 to a bag of variables in 𝐽 βŠ† 𝐼. (Contributed by SN, 10-Mar-2025.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΈ = {𝑔 ∈ (β„•0 ↑m 𝐽) ∣ (◑𝑔 β€œ β„•) ∈ Fin}    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    β‡’   (πœ‘ β†’ (𝐹 β†Ύ 𝐽) ∈ 𝐸)
 
Theoremmpllmodd 41419 The polynomial ring is a left module. (Contributed by SN, 12-Mar-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑃 ∈ LMod)
 
Theoremmplringd 41420 The polynomial ring is a ring. (Contributed by SN, 7-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑃 ∈ Ring)
 
Theoremmplcrngd 41421 The polynomial ring is a commutative ring. (Contributed by SN, 7-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑃 ∈ CRing)
 
Theoremmplsubrgcl 41422 An element of a polynomial algebra over a subring is an element of the polynomial algebra. (Contributed by SN, 9-Feb-2025.)
π‘Š = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π΅ = (Baseβ€˜π‘Š)    &   π‘ƒ = (𝐼 mPoly 𝑆)    &   πΆ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝐹 ∈ 𝐢)
 
Theoremmhmcompl 41423 The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 7-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π‘„ = (𝐼 mPoly 𝑆)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΆ = (Baseβ€˜π‘„)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐻 ∈ (𝑅 MndHom 𝑆))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐻 ∘ 𝐹) ∈ 𝐢)
 
Theoremrhmmpllem1 41424* Lemma for rhmmpl 41428. A subproof of psrmulcllem 21726. (Contributed by SN, 8-Feb-2025.)
𝐷 = {𝑓 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋:𝐷⟢(Baseβ€˜π‘…))    &   (πœ‘ β†’ π‘Œ:𝐷⟢(Baseβ€˜π‘…))    β‡’   ((πœ‘ ∧ π‘˜ ∈ 𝐷) β†’ (π‘₯ ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≀ π‘˜} ↦ ((π‘‹β€˜π‘₯)(.rβ€˜π‘…)(π‘Œβ€˜(π‘˜ ∘f βˆ’ π‘₯)))) finSupp (0gβ€˜π‘…))
 
Theoremrhmmpllem2 41425* Lemma for rhmmpl 41428. A subproof of psrmulcllem 21726. (Contributed by SN, 8-Feb-2025.)
𝐷 = {𝑓 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋:𝐷⟢(Baseβ€˜π‘…))    &   (πœ‘ β†’ π‘Œ:𝐷⟢(Baseβ€˜π‘…))    β‡’   ((πœ‘ ∧ π‘˜ ∈ 𝐷) β†’ (𝑅 Ξ£g (π‘₯ ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≀ π‘˜} ↦ ((π‘‹β€˜π‘₯)(.rβ€˜π‘…)(π‘Œβ€˜(π‘˜ ∘f βˆ’ π‘₯))))) ∈ (Baseβ€˜π‘…))
 
Theoremmhmcoaddmpl 41426 Show that the ring homomorphism in rhmmpl 41428 preserves addition. (Contributed by SN, 8-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π‘„ = (𝐼 mPoly 𝑆)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΆ = (Baseβ€˜π‘„)    &    + = (+gβ€˜π‘ƒ)    &    ✚ = (+gβ€˜π‘„)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐻 ∈ (𝑅 MndHom 𝑆))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)))
 
Theoremrhmcomulmpl 41427 Show that the ring homomorphism in rhmmpl 41428 preserves multiplication. (Contributed by SN, 8-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π‘„ = (𝐼 mPoly 𝑆)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΆ = (Baseβ€˜π‘„)    &    Β· = (.rβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘„)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐻 ∈ (𝑅 RingHom 𝑆))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐻 ∘ (𝐹 Β· 𝐺)) = ((𝐻 ∘ 𝐹) βˆ™ (𝐻 ∘ 𝐺)))
 
Theoremrhmmpl 41428* Provide a ring homomorphism between two polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. Compare pwsco2rhm 20395. TODO: Currently mhmvlin 22120 would have to be moved up. Investigate the usefulness of surrounding theorems like mndvcl 22114 and the difference between mhmvlin 22120, ofco 7696, and ofco2 22174. (Contributed by SN, 8-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π‘„ = (𝐼 mPoly 𝑆)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΉ = (𝑝 ∈ 𝐡 ↦ (𝐻 ∘ 𝑝))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐻 ∈ (𝑅 RingHom 𝑆))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑃 RingHom 𝑄))
 
Theoremmplascl0 41429 The zero scalar as a polynomial. (Contributed by SN, 23-Nov-2024.)
π‘Š = (𝐼 mPoly 𝑅)    &   π΄ = (algScβ€˜π‘Š)    &   π‘‚ = (0gβ€˜π‘…)    &    0 = (0gβ€˜π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ (π΄β€˜π‘‚) = 0 )
 
Theoremmplascl1 41430 The one scalar as a polynomial. (Contributed by SN, 12-Mar-2025.)
π‘Š = (𝐼 mPoly 𝑅)    &   π΄ = (algScβ€˜π‘Š)    &   π‘‚ = (1rβ€˜π‘…)    &    1 = (1rβ€˜π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ (π΄β€˜π‘‚) = 1 )
 
Theoremmplmapghm 41431* 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)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π· = {𝑓 ∈ (β„•0 ↑m 𝐼) ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π» = (𝑝 ∈ 𝐡 ↦ (π‘β€˜πΉ))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Grp)    &   (πœ‘ β†’ 𝐹 ∈ 𝐷)    β‡’   (πœ‘ β†’ 𝐻 ∈ (𝑃 GrpHom 𝑅))
 
Theoremevl0 41432 The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.)
𝑄 = (𝐼 eval 𝑅)    &   π΅ = (Baseβ€˜π‘…)    &   π‘Š = (𝐼 mPoly 𝑅)    &   π‘‚ = (0gβ€˜π‘…)    &    0 = (0gβ€˜π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ (π‘„β€˜ 0 ) = ((𝐡 ↑m 𝐼) Γ— {𝑂}))
 
Theoremevlscl 41433 A polynomial over the ring 𝑅 evaluates to an element in 𝑅. (Contributed by SN, 12-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑅)β€˜π‘†)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑅 β†Ύs 𝑆)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑆 ∈ (SubRingβ€˜π‘…))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΉ)β€˜π΄) ∈ 𝐾)
 
Theoremevlsval3 41434* Give a formula for the polynomial evaluation homomorphism. (Contributed by SN, 26-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΎ = (Baseβ€˜π‘†)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π‘‡ = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   π‘€ = (mulGrpβ€˜π‘‡)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘‡)    &   πΈ = (𝑝 ∈ 𝐡 ↦ (𝑇 Ξ£g (𝑏 ∈ 𝐷 ↦ ((πΉβ€˜(π‘β€˜π‘)) Β· (𝑀 Ξ£g (𝑏 ∘f ↑ 𝐺))))))    &   πΉ = (π‘₯ ∈ 𝑅 ↦ ((𝐾 ↑m 𝐼) Γ— {π‘₯}))    &   πΊ = (π‘₯ ∈ 𝐼 ↦ (π‘Ž ∈ (𝐾 ↑m 𝐼) ↦ (π‘Žβ€˜π‘₯)))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    β‡’   (πœ‘ β†’ 𝑄 = 𝐸)
 
Theoremevlsvval 41435* Give a formula for the evaluation of a polynomial. (Contributed by SN, 9-Feb-2025.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΎ = (Baseβ€˜π‘†)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π‘‡ = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   π‘€ = (mulGrpβ€˜π‘‡)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘‡)    &   πΉ = (π‘₯ ∈ 𝑅 ↦ ((𝐾 ↑m 𝐼) Γ— {π‘₯}))    &   πΊ = (π‘₯ ∈ 𝐼 ↦ (π‘Ž ∈ (𝐾 ↑m 𝐼) ↦ (π‘Žβ€˜π‘₯)))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘„β€˜π΄) = (𝑇 Ξ£g (𝑏 ∈ 𝐷 ↦ ((πΉβ€˜(π΄β€˜π‘)) Β· (𝑀 Ξ£g (𝑏 ∘f ↑ 𝐺))))))
 
Theoremevlsvvvallem 41436* Lemma for evlsvvval 41438 akin to psrbagev2 21860. (Contributed by SN, 6-Mar-2025.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΎ = (Baseβ€˜π‘†)    &   π‘€ = (mulGrpβ€˜π‘†)    &    ↑ = (.gβ€˜π‘€)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    β‡’   (πœ‘ β†’ (𝑀 Ξ£g (𝑣 ∈ 𝐼 ↦ ((π΅β€˜π‘£) ↑ (π΄β€˜π‘£)))) ∈ 𝐾)
 
Theoremevlsvvvallem2 41437* Lemma for theorems using evlsvvval 41438. (Contributed by SN, 8-Mar-2025.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘†)    &   π‘€ = (mulGrpβ€˜π‘†)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ (𝑏 ∈ 𝐷 ↦ ((πΉβ€˜π‘) Β· (𝑀 Ξ£g (𝑣 ∈ 𝐼 ↦ ((π‘β€˜π‘£) ↑ (π΄β€˜π‘£)))))) finSupp (0gβ€˜π‘†))
 
Theoremevlsvvval 41438* Give a formula for the evaluation of a polynomial given assignments from variables to values. This is the sum of the evaluations for each term (corresponding to a bag of variables), that is, the coefficient times the product of each variable raised to the corresponding power. (Contributed by SN, 5-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΎ = (Baseβ€˜π‘†)    &   π‘€ = (mulGrpβ€˜π‘†)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΉ)β€˜π΄) = (𝑆 Ξ£g (𝑏 ∈ 𝐷 ↦ ((πΉβ€˜π‘) Β· (𝑀 Ξ£g (𝑖 ∈ 𝐼 ↦ ((π‘β€˜π‘–) ↑ (π΄β€˜π‘–))))))))
 
Theoremevlsscaval 41439 Polynomial evaluation builder for a scalar. Compare evl1scad 22075. Note that scalar multiplication by 𝑋 is the same as vector multiplication by (π΄β€˜π‘‹) by asclmul1 21660. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝑋 ∈ 𝑅)    &   (πœ‘ β†’ 𝐿 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π΄β€˜π‘‹) ∈ 𝐡 ∧ ((π‘„β€˜(π΄β€˜π‘‹))β€˜πΏ) = 𝑋))
 
Theoremevlsvarval 41440 Polynomial evaluation builder for a variable. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘‰ = (𝐼 mVar π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝑋 ∈ 𝐼)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘‰β€˜π‘‹) ∈ 𝐡 ∧ ((π‘„β€˜(π‘‰β€˜π‘‹))β€˜π΄) = (π΄β€˜π‘‹)))
 
Theoremevlsbagval 41441* 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 π‘ˆ instead of 𝑆 may not be convenient. (Contributed by SN, 29-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π‘Š = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘†)    &   π‘€ = (mulGrpβ€˜π‘†)    &    ↑ = (.gβ€˜π‘€)    &    0 = (0gβ€˜π‘ˆ)    &    1 = (1rβ€˜π‘ˆ)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΉ = (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝐡, 1 , 0 ))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    β‡’   (πœ‘ β†’ (𝐹 ∈ π‘Š ∧ ((π‘„β€˜πΉ)β€˜π΄) = (𝑀 Ξ£g (𝑣 ∈ 𝐼 ↦ ((π΅β€˜π‘£) ↑ (π΄β€˜π‘£))))))
 
Theoremevlsexpval 41442 Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ (𝑀 ∈ 𝐡 ∧ ((π‘„β€˜π‘€)β€˜π΄) = 𝑉))    &    βˆ™ = (.gβ€˜(mulGrpβ€˜π‘ƒ))    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ ((𝑁 βˆ™ 𝑀) ∈ 𝐡 ∧ ((π‘„β€˜(𝑁 βˆ™ 𝑀))β€˜π΄) = (𝑁 ↑ 𝑉)))
 
Theoremevlsaddval 41443 Polynomial evaluation builder for addition. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ (𝑀 ∈ 𝐡 ∧ ((π‘„β€˜π‘€)β€˜π΄) = 𝑉))    &   (πœ‘ β†’ (𝑁 ∈ 𝐡 ∧ ((π‘„β€˜π‘)β€˜π΄) = π‘Š))    &    ✚ = (+gβ€˜π‘ƒ)    &    + = (+gβ€˜π‘†)    β‡’   (πœ‘ β†’ ((𝑀 ✚ 𝑁) ∈ 𝐡 ∧ ((π‘„β€˜(𝑀 ✚ 𝑁))β€˜π΄) = (𝑉 + π‘Š)))
 
Theoremevlsmulval 41444 Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ (𝑀 ∈ 𝐡 ∧ ((π‘„β€˜π‘€)β€˜π΄) = 𝑉))    &   (πœ‘ β†’ (𝑁 ∈ 𝐡 ∧ ((π‘„β€˜π‘)β€˜π΄) = π‘Š))    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘†)    β‡’   (πœ‘ β†’ ((𝑀 βˆ™ 𝑁) ∈ 𝐡 ∧ ((π‘„β€˜(𝑀 βˆ™ 𝑁))β€˜π΄) = (𝑉 Β· π‘Š)))
 
Theoremevlsmaprhm 41445* The function 𝐹 mapping polynomials 𝑝 to their subring evaluation at a given point 𝑋 is a ring homomorphism. Compare evls1maprhm 33045. (Contributed by SN, 12-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑅)β€˜π‘†)    &   π‘ƒ = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑅 β†Ύs 𝑆)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   πΉ = (𝑝 ∈ 𝐡 ↦ ((π‘„β€˜π‘)β€˜π΄))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑆 ∈ (SubRingβ€˜π‘…))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑃 RingHom 𝑅))
 
Theoremevlsevl 41446 Evaluation in a subring is the same as evaluation in the ring itself. (Contributed by SN, 9-Feb-2025.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π‘‚ = (𝐼 eval 𝑆)    &   π‘Š = (𝐼 mPoly π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π΅ = (Baseβ€˜π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ (π‘„β€˜πΉ) = (π‘‚β€˜πΉ))
 
Theoremevlcl 41447 A polynomial over the ring 𝑅 evaluates to an element in 𝑅. (Contributed by SN, 12-Mar-2025.)
𝑄 = (𝐼 eval 𝑅)    &   π‘ƒ = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΉ)β€˜π΄) ∈ 𝐾)
 
Theoremevlvvval 41448* 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)    &   π‘ƒ = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΎ = (Baseβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΉ)β€˜π΄) = (𝑅 Ξ£g (𝑏 ∈ 𝐷 ↦ ((πΉβ€˜π‘) Β· (𝑀 Ξ£g (𝑖 ∈ 𝐼 ↦ ((π‘β€˜π‘–) ↑ (π΄β€˜π‘–))))))))
 
Theoremevlvvvallem 41449* Lemma for theorems using evlvvval 41448. Version of evlsvvvallem2 41437 using df-evl 21856. (Contributed by SN, 11-Mar-2025.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   π‘ƒ = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ (𝑏 ∈ 𝐷 ↦ ((πΉβ€˜π‘) Β· (𝑀 Ξ£g (𝑣 ∈ 𝐼 ↦ ((π‘β€˜π‘£) ↑ (π΄β€˜π‘£)))))) finSupp (0gβ€˜π‘…))
 
Theoremevladdval 41450 Polynomial evaluation builder for addition. (Contributed by SN, 9-Feb-2025.)
𝑄 = (𝐼 eval 𝑆)    &   π‘ƒ = (𝐼 mPoly 𝑆)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &    ✚ = (+gβ€˜π‘ƒ)    &    + = (+gβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ (𝑀 ∈ 𝐡 ∧ ((π‘„β€˜π‘€)β€˜π΄) = 𝑉))    &   (πœ‘ β†’ (𝑁 ∈ 𝐡 ∧ ((π‘„β€˜π‘)β€˜π΄) = π‘Š))    β‡’   (πœ‘ β†’ ((𝑀 ✚ 𝑁) ∈ 𝐡 ∧ ((π‘„β€˜(𝑀 ✚ 𝑁))β€˜π΄) = (𝑉 + π‘Š)))
 
Theoremevlmulval 41451 Polynomial evaluation builder for multiplication. (Contributed by SN, 18-Feb-2025.)
𝑄 = (𝐼 eval 𝑆)    &   π‘ƒ = (𝐼 mPoly 𝑆)    &   πΎ = (Baseβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝑍)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (πœ‘ β†’ (𝑀 ∈ 𝐡 ∧ ((π‘„β€˜π‘€)β€˜π΄) = 𝑉))    &   (πœ‘ β†’ (𝑁 ∈ 𝐡 ∧ ((π‘„β€˜π‘)β€˜π΄) = π‘Š))    β‡’   (πœ‘ β†’ ((𝑀 βˆ™ 𝑁) ∈ 𝐡 ∧ ((π‘„β€˜(𝑀 βˆ™ 𝑁))β€˜π΄) = (𝑉 Β· π‘Š)))
 
Theoremselvcllem1 41452 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼 βˆ– 𝐽) and we have 𝐽 ∈ π‘Š instead of 𝐽 βŠ† 𝐼. TODO-SN: In practice, this "simplification" makes the lemmas harder to use. (Contributed by SN, 15-Dec-2023.)
π‘ˆ = (𝐼 mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑇 ∈ AssAlg)
 
Theoremselvcllem2 41453 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
π‘ˆ = (𝐼 mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΆ = (algScβ€˜π‘‡)    &   π· = (𝐢 ∘ (algScβ€˜π‘ˆ))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝐷 ∈ (𝑅 RingHom 𝑇))
 
Theoremselvcllem3 41454 The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
π‘ˆ = (𝐼 mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΆ = (algScβ€˜π‘‡)    &   π· = (𝐢 ∘ (algScβ€˜π‘ˆ))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ ran 𝐷 ∈ (SubRingβ€˜π‘‡))
 
Theoremselvcllemh 41455 Apply the third argument (selvcllem3 41454) to show that 𝑄 is a (ring) homomorphism. (Contributed by SN, 5-Nov-2023.)
π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΆ = (algScβ€˜π‘‡)    &   π· = (𝐢 ∘ (algScβ€˜π‘ˆ))    &   π‘„ = ((𝐼 evalSub 𝑇)β€˜ran 𝐷)    &   π‘Š = (𝐼 mPoly 𝑆)    &   π‘† = (𝑇 β†Ύs ran 𝐷)    &   π‘‹ = (𝑇 ↑s (𝐡 ↑m 𝐼))    &   π΅ = (Baseβ€˜π‘‡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    β‡’   (πœ‘ β†’ 𝑄 ∈ (π‘Š RingHom 𝑋))
 
Theoremselvcllem4 41456 The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼 βˆ– 𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΆ = (algScβ€˜π‘‡)    &   π· = (𝐢 ∘ (algScβ€˜π‘ˆ))    &   π‘† = (𝑇 β†Ύs ran 𝐷)    &   π‘Š = (𝐼 mPoly 𝑆)    &   π‘‹ = (Baseβ€˜π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐷 ∘ 𝐹) ∈ 𝑋)
 
Theoremselvcllem5 41457* The fifth argument passed to evalSub is in the domain (a function 𝐼⟢𝐸). (Contributed by SN, 22-Feb-2024.)
π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΆ = (algScβ€˜π‘‡)    &   πΈ = (Baseβ€˜π‘‡)    &   πΉ = (π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar π‘ˆ)β€˜π‘₯), (πΆβ€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝐸 ↑m 𝐼))
 
Theoremselvcl 41458 Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΈ = (Baseβ€˜π‘‡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ (((𝐼 selectVars 𝑅)β€˜π½)β€˜πΉ) ∈ 𝐸)
 
Theoremselvval2 41459* Value of the "variable selection" function. Convert selvval 21901 into a simpler form by using evlsevl 41446. (Contributed by SN, 9-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΆ = (algScβ€˜π‘‡)    &   π· = (𝐢 ∘ (algScβ€˜π‘ˆ))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    β‡’   (πœ‘ β†’ (((𝐼 selectVars 𝑅)β€˜π½)β€˜πΉ) = (((𝐼 eval 𝑇)β€˜(𝐷 ∘ 𝐹))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝐽, ((𝐽 mVar π‘ˆ)β€˜π‘₯), (πΆβ€˜(((𝐼 βˆ– 𝐽) mVar 𝑅)β€˜π‘₯))))))
 
Theoremselvvvval 41460* Recover the original polynomial from a selectVars application. (Contributed by SN, 15-Mar-2025.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   π‘ƒ = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐷)    β‡’   (πœ‘ β†’ (((((𝐼 selectVars 𝑅)β€˜π½)β€˜πΉ)β€˜(π‘Œ β†Ύ 𝐽))β€˜(π‘Œ β†Ύ (𝐼 βˆ– 𝐽))) = (πΉβ€˜π‘Œ))
 
Theoremevlselvlem 41461* Lemma for evlselv 41462. Used to re-index to and from bags of variables in 𝐼 and bags of variables in the subsets 𝐽 and 𝐼 βˆ– 𝐽. (Contributed by SN, 10-Mar-2025.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΈ = {𝑔 ∈ (β„•0 ↑m 𝐽) ∣ (◑𝑔 β€œ β„•) ∈ Fin}    &   πΆ = {𝑓 ∈ (β„•0 ↑m (𝐼 βˆ– 𝐽)) ∣ (◑𝑓 β€œ β„•) ∈ Fin}    &   π» = (𝑐 ∈ 𝐢, 𝑒 ∈ 𝐸 ↦ (𝑐 βˆͺ 𝑒))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    β‡’   (πœ‘ β†’ 𝐻:(𝐢 Γ— 𝐸)–1-1-onto→𝐷)
 
Theoremevlselv 41462 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)    &   πΎ = (Baseβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &   πΏ = (algScβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((((𝐼 βˆ– 𝐽) eval 𝑅)β€˜(((𝐽 eval π‘ˆ)β€˜(((𝐼 selectVars 𝑅)β€˜π½)β€˜πΉ))β€˜(𝐿 ∘ (𝐴 β†Ύ 𝐽))))β€˜(𝐴 β†Ύ (𝐼 βˆ– 𝐽))) = (((𝐼 eval 𝑅)β€˜πΉ)β€˜π΄))
 
Theoremselvadd 41463 The "variable selection" function is additive. (Contributed by SN, 7-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &    + = (+gβ€˜π‘ƒ)    &   π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &    ✚ = (+gβ€˜π‘‡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    β‡’   (πœ‘ β†’ (((𝐼 selectVars 𝑅)β€˜π½)β€˜(𝐹 + 𝐺)) = ((((𝐼 selectVars 𝑅)β€˜π½)β€˜πΉ) ✚ (((𝐼 selectVars 𝑅)β€˜π½)β€˜πΊ)))
 
Theoremselvmul 41464 The "variable selection" function is multiplicative. (Contributed by SN, 18-Feb-2025.)
𝑃 = (𝐼 mPoly 𝑅)    &   π΅ = (Baseβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘ƒ)    &   π‘ˆ = ((𝐼 βˆ– 𝐽) mPoly 𝑅)    &   π‘‡ = (𝐽 mPoly π‘ˆ)    &    βˆ™ = (.rβ€˜π‘‡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐽 βŠ† 𝐼)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    β‡’   (πœ‘ β†’ (((𝐼 selectVars 𝑅)β€˜π½)β€˜(𝐹 Β· 𝐺)) = ((((𝐼 selectVars 𝑅)β€˜π½)β€˜πΉ) βˆ™ (((𝐼 selectVars 𝑅)β€˜π½)β€˜πΊ)))
 
Theoremfsuppind 41465* Induction on functions 𝐹:𝐴⟢𝐡 with finite support, or in other words the base set of the free module (see frlmelbas 21531 and frlmplusgval 21539). This theorem is structurally general for polynomial proof usage (see mplelbas 21770 and mpladd 21788). Note that hypothesis 0 is redundant when 𝐼 is nonempty. (Contributed by SN, 18-May-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ (𝐼 Γ— { 0 }) ∈ 𝐻)    &   ((πœ‘ ∧ (π‘Ž ∈ 𝐼 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘₯ ∈ 𝐼 ↦ if(π‘₯ = π‘Ž, 𝑏, 0 )) ∈ 𝐻)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) β†’ (π‘₯ ∘f + 𝑦) ∈ 𝐻)    β‡’   ((πœ‘ ∧ (𝑋:𝐼⟢𝐡 ∧ 𝑋 finSupp 0 )) β†’ 𝑋 ∈ 𝐻)
 
Theoremfsuppssindlem1 41466* Lemma for fsuppssind 41468. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
(πœ‘ β†’ 0 ∈ π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐼⟢𝐡)    &   (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝑆)    β‡’   (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑆, ((𝐹 β†Ύ 𝑆)β€˜π‘₯), 0 )))
 
Theoremfsuppssindlem2 41467* Lemma for fsuppssind 41468. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
(πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐼)    β‡’   (πœ‘ β†’ (𝐹 ∈ {𝑓 ∈ (𝐡 ↑m 𝑆) ∣ (π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑆, (π‘“β€˜π‘₯), 0 )) ∈ 𝐻} ↔ (𝐹:π‘†βŸΆπ΅ ∧ (𝐹 βˆͺ ((𝐼 βˆ– 𝑆) Γ— { 0 })) ∈ 𝐻)))
 
Theoremfsuppssind 41468* Induction on functions 𝐹:𝐴⟢𝐡 with finite support (see fsuppind 41465) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)
𝐡 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 βŠ† 𝐼)    &   (πœ‘ β†’ (𝐼 Γ— { 0 }) ∈ 𝐻)    &   ((πœ‘ ∧ (π‘Ž ∈ 𝑆 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑠 ∈ 𝐼 ↦ if(𝑠 = π‘Ž, 𝑏, 0 )) ∈ 𝐻)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) β†’ (π‘₯ ∘f + 𝑦) ∈ 𝐻)    &   (πœ‘ β†’ 𝑋:𝐼⟢𝐡)    &   (πœ‘ β†’ 𝑋 finSupp 0 )    &   (πœ‘ β†’ (𝑋 supp 0 ) βŠ† 𝑆)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝐻)
 
Theoremmhpind 41469* 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)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘ƒ = (𝐼 mPoly 𝑅)    &    + = (+gβ€˜π‘ƒ)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   π‘† = {𝑔 ∈ 𝐷 ∣ ((β„‚fld β†Ύs β„•0) Ξ£g 𝑔) = 𝑁}    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅 ∈ Grp)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ (π»β€˜π‘))    &   (πœ‘ β†’ (𝐷 Γ— { 0 }) ∈ 𝐺)    &   ((πœ‘ ∧ (π‘Ž ∈ 𝑆 ∧ 𝑏 ∈ 𝐡)) β†’ (𝑠 ∈ 𝐷 ↦ if(𝑠 = π‘Ž, 𝑏, 0 )) ∈ 𝐺)    &   ((πœ‘ ∧ (π‘₯ ∈ ((π»β€˜π‘) ∩ 𝐺) ∧ 𝑦 ∈ ((π»β€˜π‘) ∩ 𝐺))) β†’ (π‘₯ + 𝑦) ∈ 𝐺)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝐺)
 
Theoremevlsmhpvvval 41470* Give a formula for the evaluation of a homogeneous polynomial given assignments from variables to values. The difference between this and evlsvvval 41438 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)β€˜π‘…)    &   π» = (𝐼 mHomP π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   π· = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   πΊ = {𝑔 ∈ 𝐷 ∣ ((β„‚fld β†Ύs β„•0) Ξ£g 𝑔) = 𝑁}    &   πΎ = (Baseβ€˜π‘†)    &   π‘€ = (mulGrpβ€˜π‘†)    &    ↑ = (.gβ€˜π‘€)    &    Β· = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐹 ∈ (π»β€˜π‘))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΉ)β€˜π΄) = (𝑆 Ξ£g (𝑏 ∈ 𝐺 ↦ ((πΉβ€˜π‘) Β· (𝑀 Ξ£g (𝑖 ∈ 𝐼 ↦ ((π‘β€˜π‘–) ↑ (π΄β€˜π‘–))))))))
 
Theoremmhphflem 41471* Lemma for mhphf 41472. Add several multiples of 𝐿 together, in a case where the total amount of multiplies is 𝑁. (Contributed by SN, 30-Jul-2024.)
𝐷 = {β„Ž ∈ (β„•0 ↑m 𝐼) ∣ (β—‘β„Ž β€œ β„•) ∈ Fin}    &   π» = {𝑔 ∈ 𝐷 ∣ ((β„‚fld β†Ύs β„•0) Ξ£g 𝑔) = 𝑁}    &   π΅ = (Baseβ€˜πΊ)    &    Β· = (.gβ€˜πΊ)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝐿 ∈ 𝐡)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   ((πœ‘ ∧ π‘Ž ∈ 𝐻) β†’ (𝐺 Ξ£g (𝑣 ∈ 𝐼 ↦ ((π‘Žβ€˜π‘£) Β· 𝐿))) = (𝑁 Β· 𝐿))
 
Theoremmhphf 41472 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 (π‘„β€˜π‘‹) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.) (Proof shortened by SN, 8-Mar-2025.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π» = (𝐼 mHomP π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &    Β· = (.rβ€˜π‘†)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐿 ∈ 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ (π»β€˜π‘))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜π‘‹)β€˜((𝐼 Γ— {𝐿}) ∘f Β· 𝐴)) = ((𝑁 ↑ 𝐿) Β· ((π‘„β€˜π‘‹)β€˜π΄)))
 
Theoremmhphf2 41473 A homogeneous polynomial defines a homogeneous function; this is mhphf 41472 with simpler notation in the conclusion in exchange for a complex definition of βˆ™, which is based on frlmvscafval 21541 but without the finite support restriction (frlmpws 21525, frlmbas 21530) on the assignments 𝐴 from variables to values.

TODO?: Polynomials (df-mpl 21684) 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)β€˜π‘…)    &   π» = (𝐼 mHomP π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &    βˆ™ = ( ·𝑠 β€˜((ringLModβ€˜π‘†) ↑s 𝐼))    &    Β· = (.rβ€˜π‘†)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐿 ∈ 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ (π»β€˜π‘))    &   (πœ‘ β†’ 𝐴 ∈ (𝐾 ↑m 𝐼))    β‡’   (πœ‘ β†’ ((π‘„β€˜π‘‹)β€˜(𝐿 βˆ™ 𝐴)) = ((𝑁 ↑ 𝐿) Β· ((π‘„β€˜π‘‹)β€˜π΄)))
 
Theoremmhphf3 41474 A homogeneous polynomial defines a homogeneous function; this is mhphf2 41473 with the finite support restriction (frlmpws 21525, frlmbas 21530) on the assignments 𝐴 from variables to values. See comment of mhphf2 41473. (Contributed by SN, 23-Nov-2024.)
𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)    &   π» = (𝐼 mHomP π‘ˆ)    &   π‘ˆ = (𝑆 β†Ύs 𝑅)    &   πΎ = (Baseβ€˜π‘†)    &   πΉ = (𝑆 freeLMod 𝐼)    &   π‘€ = (Baseβ€˜πΉ)    &    βˆ™ = ( ·𝑠 β€˜πΉ)    &    Β· = (.rβ€˜π‘†)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝑅 ∈ (SubRingβ€˜π‘†))    &   (πœ‘ β†’ 𝐿 ∈ 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ (π»β€˜π‘))    &   (πœ‘ β†’ 𝐴 ∈ 𝑀)    β‡’   (πœ‘ β†’ ((π‘„β€˜π‘‹)β€˜(𝐿 βˆ™ 𝐴)) = ((𝑁 ↑ 𝐿) Β· ((π‘„β€˜π‘‹)β€˜π΄)))
 
Theoremmhphf4 41475 A homogeneous polynomial defines a homogeneous function; this is mhphf3 41474 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.)
𝑄 = (𝐼 eval 𝑆)    &   π» = (𝐼 mHomP 𝑆)    &   πΎ = (Baseβ€˜π‘†)    &   πΉ = (𝑆 freeLMod 𝐼)    &   π‘€ = (Baseβ€˜πΉ)    &    βˆ™ = ( ·𝑠 β€˜πΉ)    &    Β· = (.rβ€˜π‘†)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ CRing)    &   (πœ‘ β†’ 𝐿 ∈ 𝐾)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ (π»β€˜π‘))    &   (πœ‘ β†’ 𝐴 ∈ 𝑀)    β‡’   (πœ‘ β†’ ((π‘„β€˜π‘‹)β€˜(𝐿 βˆ™ 𝐴)) = ((𝑁 ↑ 𝐿) Β· ((π‘„β€˜π‘‹)β€˜π΄)))
 
21.28.3  Arithmetic theorems

Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results.

For example, ax-1rid 11183 is used in mulrid 11217 related theorems, so one could trade off the extra axioms in mulrid 11217 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 11170; in the other direction, real number closure laws can be avoided by using ax-resscn 11170 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐡) only needs to be a complex number).

The natural numbers are especially amenable to axiom reductions, as the set β„• is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This conveniently allows for adding natural numbers by rearranging parentheses, as shown below:

(4 + 3) = 7

((3 + 1) + (2 + 1)) = (6 + 1)

((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) =

((((((1 + 1) + 1) + 1) + 1) + 1) + 1)

This only requires ax-addass 11178, ax-1cn 11171, and ax-addcl 11173. (And in practice, the expression isn't fully expanded into ones.)

Multiplication by 1 requires either mullidi 11224 or (ax-1rid 11183 and 1re 11219) as seen in 1t1e1 12379 and 1t1e1ALT 41479. Multiplying with greater natural numbers uses ax-distr 11180. Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as (9 + 1) = 10. Adding zero uses almost every complex number axiom, though notably not ax-mulcom 11177 (see readdrid 41585 and readdlid 41579).

 
Theoremc0exALT 41476 Alternate proof of c0ex 11213 using more set theory axioms but fewer complex number axioms (add ax-10 2136, ax-11 2153, ax-13 2370, ax-nul 5307, and remove ax-1cn 11171, ax-icn 11172, ax-addcl 11173, and ax-mulcl 11175). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ V
 
Theorem0cnALT3 41477 Alternate proof of 0cn 11211 using ax-resscn 11170, ax-addrcl 11174, ax-rnegex 11184, ax-cnre 11186 instead of ax-icn 11172, ax-addcl 11173, ax-mulcl 11175, ax-i2m1 11181. Version of 0cnALT 11453 using ax-1cn 11171 instead of ax-icn 11172. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ β„‚
 
Theoremelre0re 41478 Specialized version of 0red 11222 without using ax-1cn 11171 and ax-cnre 11186. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝐴 ∈ ℝ β†’ 0 ∈ ℝ)
 
Theorem1t1e1ALT 41479 Alternate proof of 1t1e1 12379 using a different set of axioms (add ax-mulrcl 11176, ax-i2m1 11181, ax-1ne0 11182, ax-rrecex 11185 and remove ax-resscn 11170, ax-mulcom 11177, ax-mulass 11179, ax-distr 11180). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 Β· 1) = 1
 
Theoremremulcan2d 41480 mulcan2d 11853 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 β‰  0)    β‡’   (πœ‘ β†’ ((𝐴 Β· 𝐢) = (𝐡 Β· 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremreaddridaddlidd 41481 Given some real number 𝐡 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 11393, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (𝐡 + 𝐴) = 𝐡)    β‡’   ((πœ‘ ∧ 𝐢 ∈ ℝ) β†’ (𝐴 + 𝐢) = 𝐢)
 
Theoremsn-1ne2 41482 A proof of 1ne2 12425 without using ax-mulcom 11177, ax-mulass 11179, ax-pre-mulgt0 11190. Based on mul02lem2 11396. (Contributed by SN, 13-Dec-2023.)
1 β‰  2
 
Theoremnnn1suc 41483* A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)
((𝐴 ∈ β„• ∧ 𝐴 β‰  1) β†’ βˆƒπ‘₯ ∈ β„• (π‘₯ + 1) = 𝐴)
 
Theoremnnadd1com 41484 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ β„• β†’ (𝐴 + 1) = (1 + 𝐴))
 
Theoremnnaddcom 41485 Addition is commutative for natural numbers. Uses fewer axioms than addcom 11405. (Contributed by Steven Nguyen, 9-Dec-2022.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (𝐴 + 𝐡) = (𝐡 + 𝐴))
 
Theoremnnaddcomli 41486 Version of addcomli 11411 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
𝐴 ∈ β„•    &   π΅ ∈ β„•    &   (𝐴 + 𝐡) = 𝐢    β‡’   (𝐡 + 𝐴) = 𝐢
 
Theoremnnadddir 41487 Right-distributivity for natural numbers without ax-mulcom 11177. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ 𝐢 ∈ β„•) β†’ ((𝐴 + 𝐡) Β· 𝐢) = ((𝐴 Β· 𝐢) + (𝐡 Β· 𝐢)))
 
Theoremnnmul1com 41488 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 11177. Since (𝐴 Β· 1) is 𝐴 by ax-1rid 11183, this is equivalent to remullid 41609 for natural numbers, but using fewer axioms (avoiding ax-resscn 11170, ax-addass 11178, ax-mulass 11179, ax-rnegex 11184, ax-pre-lttri 11187, ax-pre-lttrn 11188, ax-pre-ltadd 11189). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ β„• β†’ (1 Β· 𝐴) = (𝐴 Β· 1))
 
Theoremnnmulcom 41489 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„•) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
 
Theoremmvrrsubd 41490 Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11629. (Contributed by SN, 21-Aug-2024.)
(πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 = (𝐡 βˆ’ 𝐢))    β‡’   (πœ‘ β†’ (𝐴 + 𝐢) = 𝐡)
 
Theoremladdrotrd 41491 Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11630 with a commuted consequent, and of mvrladdd 11632 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (𝐴 + 𝐡) = 𝐢)    β‡’   (πœ‘ β†’ (𝐢 βˆ’ 𝐴) = 𝐡)
 
Theoremraddcom12d 41492 Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11631 with a commuted consequent, and of mvlraddd 11629 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 = (𝐡 + 𝐢))    β‡’   (πœ‘ β†’ 𝐡 = (𝐴 βˆ’ 𝐢))
 
Theoremlsubrotld 41493 Rotate the variables left in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 21-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = 𝐢)    β‡’   (πœ‘ β†’ (𝐡 + 𝐢) = 𝐴)
 
Theoremlsubcom23d 41494 Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 23-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (𝐴 βˆ’ 𝐡) = 𝐢)    β‡’   (πœ‘ β†’ (𝐴 βˆ’ 𝐢) = 𝐡)
 
Theoremaddsubeq4com 41495 Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝐢 ∈ β„‚ ∧ 𝐷 ∈ β„‚)) β†’ ((𝐴 + 𝐡) = (𝐢 + 𝐷) ↔ (𝐴 βˆ’ 𝐢) = (𝐷 βˆ’ 𝐡)))
 
Theoremsqsumi 41496 A sum squared. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    β‡’   ((𝐴 + 𝐡) Β· (𝐴 + 𝐡)) = (((𝐴 Β· 𝐴) + (𝐡 Β· 𝐡)) + (2 Β· (𝐴 Β· 𝐡)))
 
Theoremnegn0nposznnd 41497 Lemma for dffltz 41679. (Contributed by Steven Nguyen, 27-Feb-2023.)
(πœ‘ β†’ 𝐴 β‰  0)    &   (πœ‘ β†’ Β¬ 0 < 𝐴)    &   (πœ‘ β†’ 𝐴 ∈ β„€)    β‡’   (πœ‘ β†’ -𝐴 ∈ β„•)
 
Theoremsqmid3api 41498 Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.)
𝐴 ∈ β„‚    &   π‘ ∈ β„‚    &   (𝐴 + 𝑁) = 𝐡    &   (𝐡 + 𝑁) = 𝐢    β‡’   (𝐡 Β· 𝐡) = ((𝐴 Β· 𝐢) + (𝑁 Β· 𝑁))
 
Theoremdecaddcom 41499 Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
𝐴 ∈ β„•0    &   π΅ ∈ β„•0    &   πΆ ∈ β„•0    β‡’   (𝐴𝐡 + 𝐢) = (𝐴𝐢 + 𝐡)
 
Theoremsqn5i 41500 The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ β„•0    β‡’   (𝐴5 Β· 𝐴5) = (𝐴 Β· (𝐴 + 1))25
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47941
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