Theorem rlmval2 21106

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 21105	. . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
2	1	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3		ssid 3972	. . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊)
4		sraval 21089	. . 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 2730	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	ressid 17221	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊)
8	7	opeq2d 4847	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
9	8	oveq2d 7406	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
10	9	oveq1d 7405	. . 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 7405	. 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 2769	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 1540   ∈ wcel 2109   ⊆ wss 3917  ⟨cop 4598  ‘cfv 6514  (class class class)co 7390   sSet csts 17140  ndxcnx 17170  Basecbs 17186   ↾_s cress 17207  ._rcmulr 17228  Scalarcsca 17230   ·_𝑠 cvsca 17231  ·_𝑖cip 17232  subringAlg csra 21085  ringLModcrglmod 21086

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-ress 17208  df-sra 21087  df-rgmod 21088

This theorem is referenced by: (None)