Metamath Proof Explorer


Theorem issubassa3

Description: A subring that is also a subspace is a subalgebra. The key theorem is islss3 . (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Hypotheses issubassa.s 𝑆 = ( 𝑊s 𝐴 )
issubassa.l 𝐿 = ( LSubSp ‘ 𝑊 )
Assertion issubassa3 ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → 𝑆 ∈ AssAlg )

Proof

Step Hyp Ref Expression
1 issubassa.s 𝑆 = ( 𝑊s 𝐴 )
2 issubassa.l 𝐿 = ( LSubSp ‘ 𝑊 )
3 1 subrgbas ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) → 𝐴 = ( Base ‘ 𝑆 ) )
4 3 ad2antrl ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → 𝐴 = ( Base ‘ 𝑆 ) )
5 eqid ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6 1 5 resssca ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) → ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑆 ) )
7 6 ad2antrl ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑆 ) )
8 eqidd ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
9 eqid ( ·𝑠𝑊 ) = ( ·𝑠𝑊 )
10 1 9 ressvsca ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) → ( ·𝑠𝑊 ) = ( ·𝑠𝑆 ) )
11 10 ad2antrl ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( ·𝑠𝑊 ) = ( ·𝑠𝑆 ) )
12 eqid ( .r𝑊 ) = ( .r𝑊 )
13 1 12 ressmulr ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) → ( .r𝑊 ) = ( .r𝑆 ) )
14 13 ad2antrl ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( .r𝑊 ) = ( .r𝑆 ) )
15 assalmod ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod )
16 simpr ( ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) → 𝐴𝐿 )
17 1 2 lsslmod ( ( 𝑊 ∈ LMod ∧ 𝐴𝐿 ) → 𝑆 ∈ LMod )
18 15 16 17 syl2an ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → 𝑆 ∈ LMod )
19 1 subrgring ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) → 𝑆 ∈ Ring )
20 19 ad2antrl ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → 𝑆 ∈ Ring )
21 idd ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) → 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) )
22 eqid ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
23 22 subrgss ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) → 𝐴 ⊆ ( Base ‘ 𝑊 ) )
24 23 ad2antrl ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → 𝐴 ⊆ ( Base ‘ 𝑊 ) )
25 24 sseld ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( 𝑦𝐴𝑦 ∈ ( Base ‘ 𝑊 ) ) )
26 24 sseld ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( 𝑧𝐴𝑧 ∈ ( Base ‘ 𝑊 ) ) )
27 21 25 26 3anim123d ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → ( ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦𝐴𝑧𝐴 ) → ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) )
28 27 imp ( ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦𝐴𝑧𝐴 ) ) → ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) )
29 eqid ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
30 22 5 29 9 12 assaass ( ( 𝑊 ∈ AssAlg ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑥 ( ·𝑠𝑊 ) 𝑦 ) ( .r𝑊 ) 𝑧 ) = ( 𝑥 ( ·𝑠𝑊 ) ( 𝑦 ( .r𝑊 ) 𝑧 ) ) )
31 30 adantlr ( ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑥 ( ·𝑠𝑊 ) 𝑦 ) ( .r𝑊 ) 𝑧 ) = ( 𝑥 ( ·𝑠𝑊 ) ( 𝑦 ( .r𝑊 ) 𝑧 ) ) )
32 28 31 syldan ( ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦𝐴𝑧𝐴 ) ) → ( ( 𝑥 ( ·𝑠𝑊 ) 𝑦 ) ( .r𝑊 ) 𝑧 ) = ( 𝑥 ( ·𝑠𝑊 ) ( 𝑦 ( .r𝑊 ) 𝑧 ) ) )
33 22 5 29 9 12 assaassr ( ( 𝑊 ∈ AssAlg ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑦 ( .r𝑊 ) ( 𝑥 ( ·𝑠𝑊 ) 𝑧 ) ) = ( 𝑥 ( ·𝑠𝑊 ) ( 𝑦 ( .r𝑊 ) 𝑧 ) ) )
34 33 adantlr ( ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ∧ 𝑧 ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑦 ( .r𝑊 ) ( 𝑥 ( ·𝑠𝑊 ) 𝑧 ) ) = ( 𝑥 ( ·𝑠𝑊 ) ( 𝑦 ( .r𝑊 ) 𝑧 ) ) )
35 28 34 syldan ( ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝑦𝐴𝑧𝐴 ) ) → ( 𝑦 ( .r𝑊 ) ( 𝑥 ( ·𝑠𝑊 ) 𝑧 ) ) = ( 𝑥 ( ·𝑠𝑊 ) ( 𝑦 ( .r𝑊 ) 𝑧 ) ) )
36 4 7 8 11 14 18 20 32 35 isassad ( ( 𝑊 ∈ AssAlg ∧ ( 𝐴 ∈ ( SubRing ‘ 𝑊 ) ∧ 𝐴𝐿 ) ) → 𝑆 ∈ AssAlg )