ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  srascag GIF version
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 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
srapart.s (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
srapart.ex (πœ‘ β†’ π‘Š ∈ 𝑋)
Assertion
Ref Expression
srascag (πœ‘ β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
Proof of Theorem srascag
StepHypRef Expression
1 srapart.ex . . . . 5 (πœ‘ β†’ π‘Š ∈ 𝑋)
2 scaslid 12626 . . . . . . 7 (Scalar = Slot (Scalarβ€˜ndx) ∧ (Scalarβ€˜ndx) ∈ β„•)
32simpri 113 . . . . . 6 (Scalarβ€˜ndx) ∈ β„•
43a1i 9 . . . . 5 (πœ‘ β†’ (Scalarβ€˜ndx) ∈ β„•)
5 basfn 12534 . . . . . . . 8 Base Fn V
61elexd 2762 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ V)
7 funfvex 5544 . . . . . . . . 9 ((Fun Base ∧ π‘Š ∈ dom Base) β†’ (Baseβ€˜π‘Š) ∈ V)
87funfni 5328 . . . . . . . 8 ((Base Fn V ∧ π‘Š ∈ V) β†’ (Baseβ€˜π‘Š) ∈ V)
95, 6, 8sylancr 414 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘Š) ∈ V)
10 srapart.s . . . . . . 7 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
119, 10ssexd 4155 . . . . . 6 (πœ‘ β†’ 𝑆 ∈ V)
12 ressex 12539 . . . . . 6 ((π‘Š ∈ 𝑋 ∧ 𝑆 ∈ V) β†’ (π‘Š β†Ύs 𝑆) ∈ V)
131, 11, 12syl2anc 411 . . . . 5 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) ∈ V)
14 setsex 12508 . . . . 5 ((π‘Š ∈ 𝑋 ∧ (Scalarβ€˜ndx) ∈ β„• ∧ (π‘Š β†Ύs 𝑆) ∈ V) β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
151, 4, 13, 14syl3anc 1248 . . . 4 (πœ‘ β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
16 mulrslid 12605 . . . . . 6 (.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
1716slotex 12503 . . . . 5 (π‘Š ∈ 𝑋 β†’ (.rβ€˜π‘Š) ∈ V)
181, 17syl 14 . . . 4 (πœ‘ β†’ (.rβ€˜π‘Š) ∈ V)
19 vscandxnscandx 12635 . . . . . 6 ( ·𝑠 β€˜ndx) β‰  (Scalarβ€˜ndx)
2019necomi 2442 . . . . 5 (Scalarβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
21 vscaslid 12636 . . . . . 6 ( ·𝑠 = Slot ( ·𝑠 β€˜ndx) ∧ ( ·𝑠 β€˜ndx) ∈ β„•)
2221simpri 113 . . . . 5 ( ·𝑠 β€˜ndx) ∈ β„•
232, 20, 22setsslnid 12528 . . . 4 (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V ∧ (.rβ€˜π‘Š) ∈ V) β†’ (Scalarβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)) = (Scalarβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)))
2415, 18, 23syl2anc 411 . . 3 (πœ‘ β†’ (Scalarβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)) = (Scalarβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)))
2522a1i 9 . . . . 5 (πœ‘ β†’ ( ·𝑠 β€˜ndx) ∈ β„•)
26 setsex 12508 . . . . 5 (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V ∧ ( ·𝑠 β€˜ndx) ∈ β„• ∧ (.rβ€˜π‘Š) ∈ V) β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
2715, 25, 18, 26syl3anc 1248 . . . 4 (πœ‘ β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
28 slotsdifipndx 12648 . . . . . 6 (( ·𝑠 β€˜ndx) β‰  (Β·π‘–β€˜ndx) ∧ (Scalarβ€˜ndx) β‰  (Β·π‘–β€˜ndx))
2928simpri 113 . . . . 5 (Scalarβ€˜ndx) β‰  (Β·π‘–β€˜ndx)
30 ipslid 12644 . . . . . 6 (·𝑖 = Slot (Β·π‘–β€˜ndx) ∧ (Β·π‘–β€˜ndx) ∈ β„•)
3130simpri 113 . . . . 5 (Β·π‘–β€˜ndx) ∈ β„•
322, 29, 31setsslnid 12528 . . . 4 ((((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V ∧ (.rβ€˜π‘Š) ∈ V) β†’ (Scalarβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)) = (Scalarβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
3327, 18, 32syl2anc 411 . . 3 (πœ‘ β†’ (Scalarβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)) = (Scalarβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
3424, 33eqtrd 2220 . 2 (πœ‘ β†’ (Scalarβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)) = (Scalarβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
352setsslid 12527 . . 3 ((π‘Š ∈ 𝑋 ∧ (π‘Š β†Ύs 𝑆) ∈ V) β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)))
361, 13, 35syl2anc 411 . 2 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)))
37 srapart.a . . . 4 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
38 sraval 13626 . . . . 5 ((π‘Š ∈ V ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
396, 10, 38syl2anc 411 . . . 4 (πœ‘ β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
4037, 39eqtrd 2220 . . 3 (πœ‘ β†’ 𝐴 = (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩))
4140fveq2d 5531 . 2 (πœ‘ β†’ (Scalarβ€˜π΄) = (Scalarβ€˜(((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)))
4234, 36, 413eqtr4d 2230 1 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1363   ∈ wcel 2158   β‰  wne 2357  Vcvv 2749   βŠ† wss 3141  βŸ¨cop 3607   Fn wfn 5223  β€˜cfv 5228  (class class class)co 5888  β„•cn 8933  ndxcnx 12473   sSet csts 12474  Slot cslot 12475  Basecbs 12476   β†Ύs cress 12477  .rcmulr 12552  Scalarcsca 12554   ·𝑠 cvsca 12555  Β·π‘–cip 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
  Copyright terms: Public domain W3C validator