Theorem rlmval2 19961

Description: Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Assertion

Ref	Expression
rlmval2	⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))

Proof of Theorem rlmval2

Step	Hyp	Ref	Expression
1		rlmval 19958	. . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
2	1	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3		ssid 3982	. . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊)
4		sraval 19943	. . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘𝑊) ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
5	3, 4	mpan2 689	. 2 ⊢ (𝑊 ∈ 𝑋 → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
6		eqid 2820	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	ressid 16554	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊)
8	7	opeq2d 4803	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
9	8	oveq2d 7165	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
10	9	oveq1d 7164	. . 3 ⊢ (𝑊 ∈ 𝑋 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) = ((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩))
11	10	oveq1d 7164	. 2 ⊢ (𝑊 ∈ 𝑋 → (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))
12	2, 5, 11	3eqtrd 2859	1 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ⊆ wss 3929  ⟨cop 4566  ‘cfv 6348  (class class class)co 7149  ndxcnx 16475   sSet csts 16476  Basecbs 16478   ↾_s cress 16479  ._rcmulr 16561  Scalarcsca 16563   ·_𝑠 cvsca 16564  ·_𝑖cip 16565  subringAlg csra 19935  ringLModcrglmod 19936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7152  df-oprab 7153  df-mpo 7154  df-ress 16486  df-sra 19939  df-rgmod 19940

This theorem is referenced by: (None)