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Type | Label | Description |
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Statement | ||
Theorem | riccrng1 41401 | Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.) |
β’ ((π βπ π β§ π β CRing) β π β CRing) | ||
Theorem | riccrng 41402 | A ring is commutative if and only if an isomorphic ring is commutative. (Contributed by SN, 10-Jan-2025.) |
β’ (π βπ π β (π β CRing β π β CRing)) | ||
Theorem | drnginvrn0d 41403 | A multiplicative inverse in a division ring is nonzero. (recne0d 11989 analog). (Contributed by SN, 14-Aug-2024.) |
β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ πΌ = (invrβπ ) & β’ (π β π β DivRing) & β’ (π β π β π΅) & β’ (π β π β 0 ) β β’ (π β (πΌβπ) β 0 ) | ||
Theorem | drngmulcanad 41404 | Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11854 analog). (Contributed by SN, 14-Aug-2024.) |
β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ Β· = (.rβπ ) & β’ (π β π β DivRing) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β 0 ) & β’ (π β (π Β· π) = (π Β· π)) β β’ (π β π = π) | ||
Theorem | drngmulcan2ad 41405 | Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11855 analog). (Contributed by SN, 14-Aug-2024.) |
β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ Β· = (.rβπ ) & β’ (π β π β DivRing) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β 0 ) & β’ (π β (π Β· π) = (π Β· π)) β β’ (π β π = π) | ||
Theorem | drnginvmuld 41406 | Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.) |
β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ Β· = (.rβπ ) & β’ πΌ = (invrβπ ) & β’ (π β π β DivRing) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β 0 ) & β’ (π β π β 0 ) β β’ (π β (πΌβ(π Β· π)) = ((πΌβπ) Β· (πΌβπ))) | ||
Theorem | ricdrng1 41407 | A ring isomorphism maps a division ring to a division ring. (Contributed by SN, 18-Feb-2025.) |
β’ ((π βπ π β§ π β DivRing) β π β DivRing) | ||
Theorem | ricdrng 41408 | A ring is a division ring if and only if an isomorphic ring is a division ring. (Contributed by SN, 18-Feb-2025.) |
β’ (π βπ π β (π β DivRing β π β DivRing)) | ||
Theorem | ricfld 41409 | A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025.) |
β’ (π βπ π β (π β Field β π β Field)) | ||
Theorem | lvecgrp 41410 | A vector space is a group. (Contributed by SN, 28-May-2023.) |
β’ (π β LVec β π β Grp) | ||
Theorem | lvecring 41411 | The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.) |
β’ πΉ = (Scalarβπ) β β’ (π β LVec β πΉ β Ring) | ||
Theorem | frlm0vald 41412 | All coordinates of the zero vector are zero. (Contributed by SN, 14-Aug-2024.) |
β’ πΉ = (π freeLMod πΌ) & β’ 0 = (0gβπ ) & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) β β’ (π β ((0gβπΉ)βπ½) = 0 ) | ||
Theorem | frlmsnic 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.) |
β’ π = (πΎ freeLMod {πΌ}) & β’ πΉ = (π₯ β (Baseβπ) β¦ (π₯βπΌ)) β β’ ((πΎ β Ring β§ πΌ β V) β πΉ β (π LMIso (ringLModβπΎ))) | ||
Theorem | uvccl 41414 | A unit vector is a vector. (Contributed by Steven Nguyen, 16-Jul-2023.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β (πβπ½) β π΅) | ||
Theorem | uvcn0 41415 | A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ) β β’ ((π β NzRing β§ πΌ β π β§ π½ β πΌ) β (πβπ½) β 0 ) | ||
Theorem | pwselbasr 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.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ π = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π:πΌβΆπ΅) β β’ (π β π β π) | ||
Theorem | pwsgprod 41417* | Finite products in a power structure are taken componentwise. Compare pwsgsum 19892. (Contributed by SN, 30-Jul-2024.) |
β’ π = (π βs πΌ) & β’ π΅ = (Baseβπ ) & β’ 1 = (1rβπ) & β’ π = (mulGrpβπ) & β’ π = (mulGrpβπ ) & β’ (π β πΌ β π) & β’ (π β π½ β π) & β’ (π β π β CRing) & β’ ((π β§ (π₯ β πΌ β§ π¦ β π½)) β π β π΅) & β’ (π β (π¦ β π½ β¦ (π₯ β πΌ β¦ π)) finSupp 1 ) β β’ (π β (π Ξ£g (π¦ β π½ β¦ (π₯ β πΌ β¦ π))) = (π₯ β πΌ β¦ (π Ξ£g (π¦ β π½ β¦ π)))) | ||
Theorem | psrbagres 41418* | Restrict a bag of variables in πΌ to a bag of variables in π½ β πΌ. (Contributed by SN, 10-Mar-2025.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΈ = {π β (β0 βm π½) β£ (β‘π β β) β Fin} & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π·) β β’ (π β (πΉ βΎ π½) β πΈ) | ||
Theorem | mpllmodd 41419 | The polynomial ring is a left module. (Contributed by SN, 12-Mar-2025.) |
β’ π = (πΌ mPoly π ) & β’ (π β πΌ β π) & β’ (π β π β Ring) β β’ (π β π β LMod) | ||
Theorem | mplringd 41420 | The polynomial ring is a ring. (Contributed by SN, 7-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ (π β πΌ β π) & β’ (π β π β Ring) β β’ (π β π β Ring) | ||
Theorem | mplcrngd 41421 | The polynomial ring is a commutative ring. (Contributed by SN, 7-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ (π β πΌ β π) & β’ (π β π β CRing) β β’ (π β π β CRing) | ||
Theorem | mplsubrgcl 41422 | An element of a polynomial algebra over a subring is an element of the polynomial algebra. (Contributed by SN, 9-Feb-2025.) |
β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ π΅ = (Baseβπ) & β’ π = (πΌ mPoly π) & β’ πΆ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β (SubRingβπ)) & β’ (π β πΉ β π΅) β β’ (π β πΉ β πΆ) | ||
Theorem | mhmcompl 41423 | The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 7-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π» β (π MndHom π)) & β’ (π β πΉ β π΅) β β’ (π β (π» β πΉ) β πΆ) | ||
Theorem | rhmmpllem1 41424* | Lemma for rhmmpl 41428. A subproof of psrmulcllem 21726. (Contributed by SN, 8-Feb-2025.) |
β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} & β’ (π β π β Ring) & β’ (π β π:π·βΆ(Baseβπ )) & β’ (π β π:π·βΆ(Baseβπ )) β β’ ((π β§ π β π·) β (π₯ β {π¦ β π· β£ π¦ βr β€ π} β¦ ((πβπ₯)(.rβπ )(πβ(π βf β π₯)))) finSupp (0gβπ )) | ||
Theorem | rhmmpllem2 41425* | Lemma for rhmmpl 41428. A subproof of psrmulcllem 21726. (Contributed by SN, 8-Feb-2025.) |
β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} & β’ (π β π β Ring) & β’ (π β π:π·βΆ(Baseβπ )) & β’ (π β π:π·βΆ(Baseβπ )) β β’ ((π β§ π β π·) β (π Ξ£g (π₯ β {π¦ β π· β£ π¦ βr β€ π} β¦ ((πβπ₯)(.rβπ )(πβ(π βf β π₯))))) β (Baseβπ )) | ||
Theorem | mhmcoaddmpl 41426 | Show that the ring homomorphism in rhmmpl 41428 preserves addition. (Contributed by SN, 8-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ + = (+gβπ) & β’ β = (+gβπ) & β’ (π β πΌ β π) & β’ (π β π» β (π MndHom π)) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) β β’ (π β (π» β (πΉ + πΊ)) = ((π» β πΉ) β (π» β πΊ))) | ||
Theorem | rhmcomulmpl 41427 | Show that the ring homomorphism in rhmmpl 41428 preserves multiplication. (Contributed by SN, 8-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ Β· = (.rβπ) & β’ β = (.rβπ) & β’ (π β πΌ β π) & β’ (π β π» β (π RingHom π)) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) β β’ (π β (π» β (πΉ Β· πΊ)) = ((π» β πΉ) β (π» β πΊ))) | ||
Theorem | rhmmpl 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.) |
β’ π = (πΌ mPoly π ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ πΉ = (π β π΅ β¦ (π» β π)) & β’ (π β πΌ β π) & β’ (π β π» β (π RingHom π)) β β’ (π β πΉ β (π RingHom π)) | ||
Theorem | mplascl0 41429 | The zero scalar as a polynomial. (Contributed by SN, 23-Nov-2024.) |
β’ π = (πΌ mPoly π ) & β’ π΄ = (algScβπ) & β’ π = (0gβπ ) & β’ 0 = (0gβπ) & β’ (π β πΌ β π) & β’ (π β π β Ring) β β’ (π β (π΄βπ) = 0 ) | ||
Theorem | mplascl1 41430 | The one scalar as a polynomial. (Contributed by SN, 12-Mar-2025.) |
β’ π = (πΌ mPoly π ) & β’ π΄ = (algScβπ) & β’ π = (1rβπ ) & β’ 1 = (1rβπ) & β’ (π β πΌ β π) & β’ (π β π β Ring) β β’ (π β (π΄βπ) = 1 ) | ||
Theorem | mplmapghm 41431* | The function π» mapping polynomials π to their coefficient given a bag of variables πΉ is a group homomorphism. (Contributed by SN, 15-Mar-2025.) |
β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} & β’ π» = (π β π΅ β¦ (πβπΉ)) & β’ (π β πΌ β π) & β’ (π β π β Grp) & β’ (π β πΉ β π·) β β’ (π β π» β (π GrpHom π )) | ||
Theorem | evl0 41432 | The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.) |
β’ π = (πΌ eval π ) & β’ π΅ = (Baseβπ ) & β’ π = (πΌ mPoly π ) & β’ π = (0gβπ ) & β’ 0 = (0gβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) β β’ (π β (πβ 0 ) = ((π΅ βm πΌ) Γ {π})) | ||
Theorem | evlscl 41433 | A polynomial over the ring π evaluates to an element in π . (Contributed by SN, 12-Mar-2025.) |
β’ π = ((πΌ evalSub π )βπ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ )) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπΉ)βπ΄) β πΎ) | ||
Theorem | evlsval3 41434* | Give a formula for the polynomial evaluation homomorphism. (Contributed by SN, 26-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΎ = (Baseβπ) & β’ π = (π βΎs π ) & β’ π = (π βs (πΎ βm πΌ)) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ Β· = (.rβπ) & β’ πΈ = (π β π΅ β¦ (π Ξ£g (π β π· β¦ ((πΉβ(πβπ)) Β· (π Ξ£g (π βf β πΊ)))))) & β’ πΉ = (π₯ β π β¦ ((πΎ βm πΌ) Γ {π₯})) & β’ πΊ = (π₯ β πΌ β¦ (π β (πΎ βm πΌ) β¦ (πβπ₯))) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) β β’ (π β π = πΈ) | ||
Theorem | evlsvval 41435* | Give a formula for the evaluation of a polynomial. (Contributed by SN, 9-Feb-2025.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΎ = (Baseβπ) & β’ π = (π βΎs π ) & β’ π = (π βs (πΎ βm πΌ)) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ Β· = (.rβπ) & β’ πΉ = (π₯ β π β¦ ((πΎ βm πΌ) Γ {π₯})) & β’ πΊ = (π₯ β πΌ β¦ (π β (πΎ βm πΌ) β¦ (πβπ₯))) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π΄ β π΅) β β’ (π β (πβπ΄) = (π Ξ£g (π β π· β¦ ((πΉβ(π΄βπ)) Β· (π Ξ£g (π βf β πΊ)))))) | ||
Theorem | evlsvvvallem 41436* | Lemma for evlsvvval 41438 akin to psrbagev2 21860. (Contributed by SN, 6-Mar-2025.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΎ = (Baseβπ) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β π΅ β π·) β β’ (π β (π Ξ£g (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£)))) β πΎ) | ||
Theorem | evlsvvvallem2 41437* | Lemma for theorems using evlsvvval 41438. (Contributed by SN, 8-Mar-2025.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ Β· = (.rβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π£ β πΌ β¦ ((πβπ£) β (π΄βπ£)))))) finSupp (0gβπ)) | ||
Theorem | evlsvvval 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.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π΅ = (Baseβπ) & β’ π = (π βΎs π ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΎ = (Baseβπ) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ Β· = (.rβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπΉ)βπ΄) = (π Ξ£g (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π β πΌ β¦ ((πβπ) β (π΄βπ)))))))) | ||
Theorem | evlsscaval 41439 | Polynomial evaluation builder for a scalar. Compare evl1scad 22075. Note that scalar multiplication by π is the same as vector multiplication by (π΄βπ) by asclmul1 21660. (Contributed by SN, 27-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ π΄ = (algScβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π β π ) & β’ (π β πΏ β (πΎ βm πΌ)) β β’ (π β ((π΄βπ) β π΅ β§ ((πβ(π΄βπ))βπΏ) = π)) | ||
Theorem | evlsvarval 41440 | Polynomial evaluation builder for a variable. (Contributed by SN, 27-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π = (πΌ mVar π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π β πΌ) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπ) β π΅ β§ ((πβ(πβπ))βπ΄) = (π΄βπ))) | ||
Theorem | evlsbagval 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 instead of π may not be convenient. (Contributed by SN, 29-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ π = (Baseβπ) & β’ πΎ = (Baseβπ) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ 0 = (0gβπ) & β’ 1 = (1rβπ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΉ = (π β π· β¦ if(π = π΅, 1 , 0 )) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β π΅ β π·) β β’ (π β (πΉ β π β§ ((πβπΉ)βπ΄) = (π Ξ£g (π£ β πΌ β¦ ((π΅βπ£) β (π΄βπ£)))))) | ||
Theorem | evlsexpval 41442 | Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ β = (.gβ(mulGrpβπ)) & β’ β = (.gβ(mulGrpβπ)) & β’ (π β π β β0) β β’ (π β ((π β π) β π΅ β§ ((πβ(π β π))βπ΄) = (π β π))) | ||
Theorem | evlsaddval 41443 | Polynomial evaluation builder for addition. (Contributed by SN, 27-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ β = (+gβπ) & β’ + = (+gβπ) β β’ (π β ((π β π) β π΅ β§ ((πβ(π β π))βπ΄) = (π + π))) | ||
Theorem | evlsmulval 41444 | Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ β = (.rβπ) & β’ Β· = (.rβπ) β β’ (π β ((π β π) β π΅ β§ ((πβ(π β π))βπ΄) = (π Β· π))) | ||
Theorem | evlsmaprhm 41445* | The function πΉ mapping polynomials π to their subring evaluation at a given point π is a ring homomorphism. Compare evls1maprhm 33045. (Contributed by SN, 12-Mar-2025.) |
β’ π = ((πΌ evalSub π )βπ) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ πΉ = (π β π΅ β¦ ((πβπ)βπ΄)) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ )) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β πΉ β (π RingHom π )) | ||
Theorem | evlsevl 41446 | Evaluation in a subring is the same as evaluation in the ring itself. (Contributed by SN, 9-Feb-2025.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π = (πΌ eval π) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs π ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β πΉ β π΅) β β’ (π β (πβπΉ) = (πβπΉ)) | ||
Theorem | evlcl 41447 | A polynomial over the ring π evaluates to an element in π . (Contributed by SN, 12-Mar-2025.) |
β’ π = (πΌ eval π ) & β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπΉ)βπ΄) β πΎ) | ||
Theorem | evlvvval 41448* | Give a formula for the evaluation of a polynomial given assignments from variables to values. (Contributed by SN, 5-Mar-2025.) |
β’ π = (πΌ eval π ) & β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΎ = (Baseβπ ) & β’ π = (mulGrpβπ ) & β’ β = (.gβπ) & β’ Β· = (.rβπ ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπΉ)βπ΄) = (π Ξ£g (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π β πΌ β¦ ((πβπ) β (π΄βπ)))))))) | ||
Theorem | evlvvvallem 41449* | Lemma for theorems using evlvvval 41448. Version of evlsvvvallem2 41437 using df-evl 21856. (Contributed by SN, 11-Mar-2025.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ π = (mulGrpβπ ) & β’ β = (.gβπ) & β’ Β· = (.rβπ ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β (π β π· β¦ ((πΉβπ) Β· (π Ξ£g (π£ β πΌ β¦ ((πβπ£) β (π΄βπ£)))))) finSupp (0gβπ )) | ||
Theorem | evladdval 41450 | Polynomial evaluation builder for addition. (Contributed by SN, 9-Feb-2025.) |
β’ π = (πΌ eval π) & β’ π = (πΌ mPoly π) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β = (+gβπ) & β’ + = (+gβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) β β’ (π β ((π β π) β π΅ β§ ((πβ(π β π))βπ΄) = (π + π))) | ||
Theorem | evlmulval 41451 | Polynomial evaluation builder for multiplication. (Contributed by SN, 18-Feb-2025.) |
β’ π = (πΌ eval π) & β’ π = (πΌ mPoly π) & β’ πΎ = (Baseβπ) & β’ π΅ = (Baseβπ) & β’ β = (.rβπ) & β’ Β· = (.rβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π΄ β (πΎ βm πΌ)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) & β’ (π β (π β π΅ β§ ((πβπ)βπ΄) = π)) β β’ (π β ((π β π) β π΅ β§ ((πβ(π β π))βπ΄) = (π Β· π))) | ||
Theorem | selvcllem1 41452 | π is an associative algebra. For simplicity, πΌ stands for (πΌ β π½) and we have π½ β π instead of π½ β πΌ. TODO-SN: In practice, this "simplification" makes the lemmas harder to use. (Contributed by SN, 15-Dec-2023.) |
β’ π = (πΌ mPoly π ) & β’ π = (π½ mPoly π) & β’ (π β πΌ β π) & β’ (π β π½ β π) & β’ (π β π β CRing) β β’ (π β π β AssAlg) | ||
Theorem | selvcllem2 41453 | π· is a ring homomorphism. (Contributed by SN, 15-Dec-2023.) |
β’ π = (πΌ mPoly π ) & β’ π = (π½ mPoly π) & β’ πΆ = (algScβπ) & β’ π· = (πΆ β (algScβπ)) & β’ (π β πΌ β π) & β’ (π β π½ β π) & β’ (π β π β CRing) β β’ (π β π· β (π RingHom π)) | ||
Theorem | selvcllem3 41454 | The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.) |
β’ π = (πΌ mPoly π ) & β’ π = (π½ mPoly π) & β’ πΆ = (algScβπ) & β’ π· = (πΆ β (algScβπ)) & β’ (π β πΌ β π) & β’ (π β π½ β π) & β’ (π β π β CRing) β β’ (π β ran π· β (SubRingβπ)) | ||
Theorem | selvcllemh 41455 | Apply the third argument (selvcllem3 41454) to show that π is a (ring) homomorphism. (Contributed by SN, 5-Nov-2023.) |
β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ πΆ = (algScβπ) & β’ π· = (πΆ β (algScβπ)) & β’ π = ((πΌ evalSub π)βran π·) & β’ π = (πΌ mPoly π) & β’ π = (π βΎs ran π·) & β’ π = (π βs (π΅ βm πΌ)) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) β β’ (π β π β (π RingHom π)) | ||
Theorem | selvcllem4 41456 | The fourth argument passed to evalSub is in the domain (a polynomial in (πΌ mPoly (π½ mPoly ((πΌ β π½) mPoly π )))). (Contributed by SN, 5-Nov-2023.) |
β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ πΆ = (algScβπ) & β’ π· = (πΆ β (algScβπ)) & β’ π = (π βΎs ran π·) & β’ π = (πΌ mPoly π) & β’ π = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) β β’ (π β (π· β πΉ) β π) | ||
Theorem | selvcllem5 41457* | The fifth argument passed to evalSub is in the domain (a function πΌβΆπΈ). (Contributed by SN, 22-Feb-2024.) |
β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ πΆ = (algScβπ) & β’ πΈ = (Baseβπ) & β’ πΉ = (π₯ β πΌ β¦ if(π₯ β π½, ((π½ mVar π)βπ₯), (πΆβ(((πΌ β π½) mVar π )βπ₯)))) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) β β’ (π β πΉ β (πΈ βm πΌ)) | ||
Theorem | selvcl 41458 | Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.) |
β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ πΈ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) β β’ (π β (((πΌ selectVars π )βπ½)βπΉ) β πΈ) | ||
Theorem | selvval2 41459* | Value of the "variable selection" function. Convert selvval 21901 into a simpler form by using evlsevl 41446. (Contributed by SN, 9-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ πΆ = (algScβπ) & β’ π· = (πΆ β (algScβπ)) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) β β’ (π β (((πΌ selectVars π )βπ½)βπΉ) = (((πΌ eval π)β(π· β πΉ))β(π₯ β πΌ β¦ if(π₯ β π½, ((π½ mVar π)βπ₯), (πΆβ(((πΌ β π½) mVar π )βπ₯)))))) | ||
Theorem | selvvvval 41460* | Recover the original polynomial from a selectVars application. (Contributed by SN, 15-Mar-2025.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) & β’ (π β π β π·) β β’ (π β (((((πΌ selectVars π )βπ½)βπΉ)β(π βΎ π½))β(π βΎ (πΌ β π½))) = (πΉβπ)) | ||
Theorem | evlselvlem 41461* | Lemma for evlselv 41462. Used to re-index to and from bags of variables in πΌ and bags of variables in the subsets π½ and πΌ β π½. (Contributed by SN, 10-Mar-2025.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΈ = {π β (β0 βm π½) β£ (β‘π β β) β Fin} & β’ πΆ = {π β (β0 βm (πΌ β π½)) β£ (β‘π β β) β Fin} & β’ π» = (π β πΆ, π β πΈ β¦ (π βͺ π)) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) β β’ (π β π»:(πΆ Γ πΈ)β1-1-ontoβπ·) | ||
Theorem | evlselv 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.) |
β’ π = (πΌ mPoly π ) & β’ πΎ = (Baseβπ ) & β’ π΅ = (Baseβπ) & β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ πΏ = (algScβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((((πΌ β π½) eval π )β(((π½ eval π)β(((πΌ selectVars π )βπ½)βπΉ))β(πΏ β (π΄ βΎ π½))))β(π΄ βΎ (πΌ β π½))) = (((πΌ eval π )βπΉ)βπ΄)) | ||
Theorem | selvadd 41463 | The "variable selection" function is additive. (Contributed by SN, 7-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ + = (+gβπ) & β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ β = (+gβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) β β’ (π β (((πΌ selectVars π )βπ½)β(πΉ + πΊ)) = ((((πΌ selectVars π )βπ½)βπΉ) β (((πΌ selectVars π )βπ½)βπΊ))) | ||
Theorem | selvmul 41464 | The "variable selection" function is multiplicative. (Contributed by SN, 18-Feb-2025.) |
β’ π = (πΌ mPoly π ) & β’ π΅ = (Baseβπ) & β’ Β· = (.rβπ) & β’ π = ((πΌ β π½) mPoly π ) & β’ π = (π½ mPoly π) & β’ β = (.rβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π½ β πΌ) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) β β’ (π β (((πΌ selectVars π )βπ½)β(πΉ Β· πΊ)) = ((((πΌ selectVars π )βπ½)βπΉ) β (((πΌ selectVars π )βπ½)βπΊ))) | ||
Theorem | fsuppind 41465* | Induction on functions πΉ:π΄βΆπ΅ 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 πΌ is nonempty. (Contributed by SN, 18-May-2024.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ + = (+gβπΊ) & β’ (π β πΊ β Grp) & β’ (π β πΌ β π) & β’ (π β (πΌ Γ { 0 }) β π») & β’ ((π β§ (π β πΌ β§ π β π΅)) β (π₯ β πΌ β¦ if(π₯ = π, π, 0 )) β π») & β’ ((π β§ (π₯ β π» β§ π¦ β π»)) β (π₯ βf + π¦) β π») β β’ ((π β§ (π:πΌβΆπ΅ β§ π finSupp 0 )) β π β π») | ||
Theorem | fsuppssindlem1 41466* | Lemma for fsuppssind 41468. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.) |
β’ (π β 0 β π) & β’ (π β πΌ β π) & β’ (π β πΉ:πΌβΆπ΅) & β’ (π β (πΉ supp 0 ) β π) β β’ (π β πΉ = (π₯ β πΌ β¦ if(π₯ β π, ((πΉ βΎ π)βπ₯), 0 ))) | ||
Theorem | fsuppssindlem2 41467* | Lemma for fsuppssind 41468. Write a function as a union. (Contributed by SN, 15-Jul-2024.) |
β’ (π β π΅ β π) & β’ (π β πΌ β π) & β’ (π β π β πΌ) β β’ (π β (πΉ β {π β (π΅ βm π) β£ (π₯ β πΌ β¦ if(π₯ β π, (πβπ₯), 0 )) β π»} β (πΉ:πβΆπ΅ β§ (πΉ βͺ ((πΌ β π) Γ { 0 })) β π»))) | ||
Theorem | fsuppssind 41468* | Induction on functions πΉ:π΄βΆπ΅ with finite support (see fsuppind 41465) whose supports are subsets of π. (Contributed by SN, 15-Jun-2024.) |
β’ π΅ = (BaseβπΊ) & β’ 0 = (0gβπΊ) & β’ + = (+gβπΊ) & β’ (π β πΊ β Grp) & β’ (π β πΌ β π) & β’ (π β π β πΌ) & β’ (π β (πΌ Γ { 0 }) β π») & β’ ((π β§ (π β π β§ π β π΅)) β (π β πΌ β¦ if(π = π, π, 0 )) β π») & β’ ((π β§ (π₯ β π» β§ π¦ β π»)) β (π₯ βf + π¦) β π») & β’ (π β π:πΌβΆπ΅) & β’ (π β π finSupp 0 ) & β’ (π β (π supp 0 ) β π) β β’ (π β π β π») | ||
Theorem | mhpind 41469* | The homogeneous polynomials of degree π are generated by the terms of degree π and addition. (Contributed by SN, 28-Jul-2024.) |
β’ π» = (πΌ mHomP π ) & β’ π΅ = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π = (πΌ mPoly π ) & β’ + = (+gβπ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ π = {π β π· β£ ((βfld βΎs β0) Ξ£g π) = π} & β’ (π β πΌ β π) & β’ (π β π β Grp) & β’ (π β π β β0) & β’ (π β π β (π»βπ)) & β’ (π β (π· Γ { 0 }) β πΊ) & β’ ((π β§ (π β π β§ π β π΅)) β (π β π· β¦ if(π = π, π, 0 )) β πΊ) & β’ ((π β§ (π₯ β ((π»βπ) β© πΊ) β§ π¦ β ((π»βπ) β© πΊ))) β (π₯ + π¦) β πΊ) β β’ (π β π β πΊ) | ||
Theorem | evlsmhpvvval 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 π β π· is restricted to π β πΊ, that is, we can evaluate an π-th degree homogeneous polynomial over just the terms where the sum of all variable degrees is π. (Contributed by SN, 5-Mar-2025.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π» = (πΌ mHomP π) & β’ π = (π βΎs π ) & β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ πΊ = {π β π· β£ ((βfld βΎs β0) Ξ£g π) = π} & β’ πΎ = (Baseβπ) & β’ π = (mulGrpβπ) & β’ β = (.gβπ) & β’ Β· = (.rβπ) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β π β β0) & β’ (π β πΉ β (π»βπ)) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπΉ)βπ΄) = (π Ξ£g (π β πΊ β¦ ((πΉβπ) Β· (π Ξ£g (π β πΌ β¦ ((πβπ) β (π΄βπ)))))))) | ||
Theorem | mhphflem 41471* | Lemma for mhphf 41472. Add several multiples of πΏ together, in a case where the total amount of multiplies is π. (Contributed by SN, 30-Jul-2024.) |
β’ π· = {β β (β0 βm πΌ) β£ (β‘β β β) β Fin} & β’ π» = {π β π· β£ ((βfld βΎs β0) Ξ£g π) = π} & β’ π΅ = (BaseβπΊ) & β’ Β· = (.gβπΊ) & β’ (π β πΌ β π) & β’ (π β πΊ β Mnd) & β’ (π β πΏ β π΅) & β’ (π β π β β0) β β’ ((π β§ π β π») β (πΊ Ξ£g (π£ β πΌ β¦ ((πβπ£) Β· πΏ))) = (π Β· πΏ)) | ||
Theorem | mhphf 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 (πβπ) which corresponds to π). (Contributed by SN, 28-Jul-2024.) (Proof shortened by SN, 8-Mar-2025.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π» = (πΌ mHomP π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ Β· = (.rβπ) & β’ β = (.gβ(mulGrpβπ)) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β πΏ β π ) & β’ (π β π β β0) & β’ (π β π β (π»βπ)) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπ)β((πΌ Γ {πΏ}) βf Β· π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) | ||
Theorem | mhphf2 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 β, which is
based on frlmvscafval 21541 but without the
finite support restriction (frlmpws 21525, frlmbas 21530) on the assignments
π΄ 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.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π» = (πΌ mHomP π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ β = ( Β·π β((ringLModβπ) βs πΌ)) & β’ Β· = (.rβπ) & β’ β = (.gβ(mulGrpβπ)) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β πΏ β π ) & β’ (π β π β β0) & β’ (π β π β (π»βπ)) & β’ (π β π΄ β (πΎ βm πΌ)) β β’ (π β ((πβπ)β(πΏ β π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) | ||
Theorem | mhphf3 41474 | A homogeneous polynomial defines a homogeneous function; this is mhphf2 41473 with the finite support restriction (frlmpws 21525, frlmbas 21530) on the assignments π΄ from variables to values. See comment of mhphf2 41473. (Contributed by SN, 23-Nov-2024.) |
β’ π = ((πΌ evalSub π)βπ ) & β’ π» = (πΌ mHomP π) & β’ π = (π βΎs π ) & β’ πΎ = (Baseβπ) & β’ πΉ = (π freeLMod πΌ) & β’ π = (BaseβπΉ) & β’ β = ( Β·π βπΉ) & β’ Β· = (.rβπ) & β’ β = (.gβ(mulGrpβπ)) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β π β (SubRingβπ)) & β’ (π β πΏ β π ) & β’ (π β π β β0) & β’ (π β π β (π»βπ)) & β’ (π β π΄ β π) β β’ (π β ((πβπ)β(πΏ β π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) | ||
Theorem | mhphf4 41475 | A homogeneous polynomial defines a homogeneous function; this is mhphf3 41474 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.) |
β’ π = (πΌ eval π) & β’ π» = (πΌ mHomP π) & β’ πΎ = (Baseβπ) & β’ πΉ = (π freeLMod πΌ) & β’ π = (BaseβπΉ) & β’ β = ( Β·π βπΉ) & β’ Β· = (.rβπ) & β’ β = (.gβ(mulGrpβπ)) & β’ (π β πΌ β π) & β’ (π β π β CRing) & β’ (π β πΏ β πΎ) & β’ (π β π β β0) & β’ (π β π β (π»βπ)) & β’ (π β π΄ β π) β β’ (π β ((πβπ)β(πΏ β π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) | ||
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 (π΄ + π΅) 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). | ||
Theorem | c0exALT 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.) |
β’ 0 β V | ||
Theorem | 0cnALT3 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.) |
β’ 0 β β | ||
Theorem | elre0re 41478 | Specialized version of 0red 11222 without using ax-1cn 11171 and ax-cnre 11186. (Contributed by Steven Nguyen, 28-Jan-2023.) |
β’ (π΄ β β β 0 β β) | ||
Theorem | 1t1e1ALT 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.) |
β’ (1 Β· 1) = 1 | ||
Theorem | remulcan2d 41480 | mulcan2d 11853 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β πΆ β 0) β β’ (π β ((π΄ Β· πΆ) = (π΅ Β· πΆ) β π΄ = π΅)) | ||
Theorem | readdridaddlidd 41481 | Given some real number π΅ where π΄ acts like a right additive identity, derive that π΄ is a left additive identity. Note that the hypothesis is weaker than proving that π΄ is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 11393, π΄ is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (π΅ + π΄) = π΅) β β’ ((π β§ πΆ β β) β (π΄ + πΆ) = πΆ) | ||
Theorem | sn-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.) |
β’ 1 β 2 | ||
Theorem | nnn1suc 41483* | A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.) |
β’ ((π΄ β β β§ π΄ β 1) β βπ₯ β β (π₯ + 1) = π΄) | ||
Theorem | nnadd1com 41484 | Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.) |
β’ (π΄ β β β (π΄ + 1) = (1 + π΄)) | ||
Theorem | nnaddcom 41485 | Addition is commutative for natural numbers. Uses fewer axioms than addcom 11405. (Contributed by Steven Nguyen, 9-Dec-2022.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) = (π΅ + π΄)) | ||
Theorem | nnaddcomli 41486 | Version of addcomli 11411 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.) |
β’ π΄ β β & β’ π΅ β β & β’ (π΄ + π΅) = πΆ β β’ (π΅ + π΄) = πΆ | ||
Theorem | nnadddir 41487 | Right-distributivity for natural numbers without ax-mulcom 11177. (Contributed by SN, 5-Feb-2024.) |
β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ + π΅) Β· πΆ) = ((π΄ Β· πΆ) + (π΅ Β· πΆ))) | ||
Theorem | nnmul1com 41488 | Multiplication with 1 is commutative for natural numbers, without ax-mulcom 11177. Since (π΄ Β· 1) is π΄ 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.) |
β’ (π΄ β β β (1 Β· π΄) = (π΄ Β· 1)) | ||
Theorem | nnmulcom 41489 | Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.) |
β’ ((π΄ β β β§ π΅ β β) β (π΄ Β· π΅) = (π΅ Β· π΄)) | ||
Theorem | mvrrsubd 41490 | Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11629. (Contributed by SN, 21-Aug-2024.) |
β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π΄ = (π΅ β πΆ)) β β’ (π β (π΄ + πΆ) = π΅) | ||
Theorem | laddrotrd 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.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (π΄ + π΅) = πΆ) β β’ (π β (πΆ β π΄) = π΅) | ||
Theorem | raddcom12d 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.) |
β’ (π β π΅ β β) & β’ (π β πΆ β β) & β’ (π β π΄ = (π΅ + πΆ)) β β’ (π β π΅ = (π΄ β πΆ)) | ||
Theorem | lsubrotld 41493 | Rotate the variables left in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 21-Aug-2024.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (π΄ β π΅) = πΆ) β β’ (π β (π΅ + πΆ) = π΄) | ||
Theorem | lsubcom23d 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.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β (π΄ β π΅) = πΆ) β β’ (π β (π΄ β πΆ) = π΅) | ||
Theorem | addsubeq4com 41495 | Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.) |
β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((π΄ + π΅) = (πΆ + π·) β (π΄ β πΆ) = (π· β π΅))) | ||
Theorem | sqsumi 41496 | A sum squared. (Contributed by Steven Nguyen, 16-Sep-2022.) |
β’ π΄ β β & β’ π΅ β β β β’ ((π΄ + π΅) Β· (π΄ + π΅)) = (((π΄ Β· π΄) + (π΅ Β· π΅)) + (2 Β· (π΄ Β· π΅))) | ||
Theorem | negn0nposznnd 41497 | Lemma for dffltz 41679. (Contributed by Steven Nguyen, 27-Feb-2023.) |
β’ (π β π΄ β 0) & β’ (π β Β¬ 0 < π΄) & β’ (π β π΄ β β€) β β’ (π β -π΄ β β) | ||
Theorem | sqmid3api 41498 | Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.) |
β’ π΄ β β & β’ π β β & β’ (π΄ + π) = π΅ & β’ (π΅ + π) = πΆ β β’ (π΅ Β· π΅) = ((π΄ Β· πΆ) + (π Β· π)) | ||
Theorem | decaddcom 41499 | Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.) |
β’ π΄ β β0 & β’ π΅ β β0 & β’ πΆ β β0 β β’ (;π΄π΅ + πΆ) = (;π΄πΆ + π΅) | ||
Theorem | sqn5i 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.) |
β’ π΄ β β0 β β’ (;π΄5 Β· ;π΄5) = ;;(π΄ Β· (π΄ + 1))25 |
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