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Theorem srascag 13631
Description: The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.)
Hypotheses
Ref Expression
srapart.a (𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))
srapart.s (𝜑𝑆 ⊆ (Base‘𝑊))
srapart.ex (𝜑𝑊𝑋)
Assertion
Ref Expression
srascag (𝜑 → (𝑊s 𝑆) = (Scalar‘𝐴))

Proof of Theorem srascag
StepHypRef Expression
1 srapart.ex . . . . 5 (𝜑𝑊𝑋)
2 scaslid 12626 . . . . . . 7 (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
32simpri 113 . . . . . 6 (Scalar‘ndx) ∈ ℕ
43a1i 9 . . . . 5 (𝜑 → (Scalar‘ndx) ∈ ℕ)
5 basfn 12534 . . . . . . . 8 Base Fn V
61elexd 2762 . . . . . . . 8 (𝜑𝑊 ∈ V)
7 funfvex 5544 . . . . . . . . 9 ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
87funfni 5328 . . . . . . . 8 ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
95, 6, 8sylancr 414 . . . . . . 7 (𝜑 → (Base‘𝑊) ∈ V)
10 srapart.s . . . . . . 7 (𝜑𝑆 ⊆ (Base‘𝑊))
119, 10ssexd 4155 . . . . . 6 (𝜑𝑆 ∈ V)
12 ressex 12539 . . . . . 6 ((𝑊𝑋𝑆 ∈ V) → (𝑊s 𝑆) ∈ V)
131, 11, 12syl2anc 411 . . . . 5 (𝜑 → (𝑊s 𝑆) ∈ V)
14 setsex 12508 . . . . 5 ((𝑊𝑋 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V)
151, 4, 13, 14syl3anc 1248 . . . 4 (𝜑 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V)
16 mulrslid 12605 . . . . . 6 (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
1716slotex 12503 . . . . 5 (𝑊𝑋 → (.r𝑊) ∈ V)
181, 17syl 14 . . . 4 (𝜑 → (.r𝑊) ∈ V)
19 vscandxnscandx 12635 . . . . . 6 ( ·𝑠 ‘ndx) ≠ (Scalar‘ndx)
2019necomi 2442 . . . . 5 (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx)
21 vscaslid 12636 . . . . . 6 ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
2221simpri 113 . . . . 5 ( ·𝑠 ‘ndx) ∈ ℕ
232, 20, 22setsslnid 12528 . . . 4 (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V ∧ (.r𝑊) ∈ V) → (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)) = (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)))
2415, 18, 23syl2anc 411 . . 3 (𝜑 → (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)) = (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)))
2522a1i 9 . . . . 5 (𝜑 → ( ·𝑠 ‘ndx) ∈ ℕ)
26 setsex 12508 . . . . 5 (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V ∧ ( ·𝑠 ‘ndx) ∈ ℕ ∧ (.r𝑊) ∈ V) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) ∈ V)
2715, 25, 18, 26syl3anc 1248 . . . 4 (𝜑 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) ∈ V)
28 slotsdifipndx 12648 . . . . . 6 (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ∧ (Scalar‘ndx) ≠ (·𝑖‘ndx))
2928simpri 113 . . . . 5 (Scalar‘ndx) ≠ (·𝑖‘ndx)
30 ipslid 12644 . . . . . 6 (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
3130simpri 113 . . . . 5 (·𝑖‘ndx) ∈ ℕ
322, 29, 31setsslnid 12528 . . . 4 ((((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) ∈ V ∧ (.r𝑊) ∈ V) → (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
3327, 18, 32syl2anc 411 . . 3 (𝜑 → (Scalar‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
3424, 33eqtrd 2220 . 2 (𝜑 → (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
352setsslid 12527 . . 3 ((𝑊𝑋 ∧ (𝑊s 𝑆) ∈ V) → (𝑊s 𝑆) = (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)))
361, 13, 35syl2anc 411 . 2 (𝜑 → (𝑊s 𝑆) = (Scalar‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)))
37 srapart.a . . . 4 (𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))
38 sraval 13626 . . . . 5 ((𝑊 ∈ V ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
396, 10, 38syl2anc 411 . . . 4 (𝜑 → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
4037, 39eqtrd 2220 . . 3 (𝜑𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
4140fveq2d 5531 . 2 (𝜑 → (Scalar‘𝐴) = (Scalar‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
4234, 36, 413eqtr4d 2230 1 (𝜑 → (𝑊s 𝑆) = (Scalar‘𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  wne 2357  Vcvv 2749  wss 3141  cop 3607   Fn wfn 5223  cfv 5228  (class class class)co 5888  cn 8933  ndxcnx 12473   sSet csts 12474  Slot cslot 12475  Basecbs 12476  s cress 12477  .rcmulr 12552  Scalarcsca 12554   ·𝑠 cvsca 12555  ·𝑖cip 12556  subringAlg csra 13622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-ltadd 7941
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-5 8995  df-6 8996  df-7 8997  df-8 8998  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-mulr 12565  df-sca 12567  df-vsca 12568  df-ip 12569  df-sra 13624
This theorem is referenced by:  sralmod  13639  rlmscabas  13649
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