Theorem rlmval2 21156

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 21155	. . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
2	1	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3		ssid 3958	. . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊)
4		sraval 21139	. . 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 692	. 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 2737	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	ressid 17183	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊)
8	7	opeq2d 4838	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
9	8	oveq2d 7384	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
10	9	oveq1d 7383	. . 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 7383	. 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 2776	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 2114   ⊆ wss 3903  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368   sSet csts 17102  ndxcnx 17132  Basecbs 17148   ↾_s cress 17169  ._rcmulr 17190  Scalarcsca 17192   ·_𝑠 cvsca 17193  ·_𝑖cip 17194  subringAlg csra 21135  ringLModcrglmod 21136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-ress 17170  df-sra 21137  df-rgmod 21138

This theorem is referenced by: (None)