Theorem rlmval2 21376

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 21375	. . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))
2	1	a1i 11	. 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)))
3		ssid 3953	. . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊)
4		sraval 21359	. . 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 704	. 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 2760	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
7	6	ressid 17336	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊)
8	7	opeq2d 4840	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩ = ⟨(Scalar‘ndx), 𝑊⟩)
9	8	oveq2d 7429	. . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))⟩) = (𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩))
10	9	oveq1d 7428	. . 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 7428	. 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 2799	1 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r‘𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r‘𝑊)⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ⟨cop 4590  ‘cfv 6533  (class class class)co 7413   sSet csts 17255  ndxcnx 17285  Basecbs 17301   ↾_s cress 17322  ._rcmulr 17343  Scalarcsca 17345   ·_𝑠 cvsca 17346  ·_𝑖cip 17347  subringAlg csra 21355  ringLModcrglmod 21356

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-ress 17323  df-sra 21357  df-rgmod 21358

This theorem is used by: (None)