Theorem rlmval2 20078

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 20075	. . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
2	1	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3		ssid 3897	. . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊)
4		sraval 20060	. . 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 691	. 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 2738	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	ressid 16655	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊)
8	7	opeq2d 4765	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
9	8	oveq2d 7180	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
10	9	oveq1d 7179	. . 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 7179	. 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 2777	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 1542   ∈ wcel 2113   ⊆ wss 3841  ⟨cop 4519  ‘cfv 6333  (class class class)co 7164  ndxcnx 16576   sSet csts 16577  Basecbs 16579   ↾_s cress 16580  ._rcmulr 16662  Scalarcsca 16664   ·_𝑠 cvsca 16665  ·_𝑖cip 16666  subringAlg csra 20052  ringLModcrglmod 20053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-ress 16587  df-sra 20056  df-rgmod 20057

This theorem is referenced by: (None)