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Theorem srascag 13567
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 12614 . . . . . . 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 12523 . . . . . . . 8 Base Fn V
61elexd 2752 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ V)
7 funfvex 5534 . . . . . . . . 9 ((Fun Base ∧ π‘Š ∈ dom Base) β†’ (Baseβ€˜π‘Š) ∈ V)
87funfni 5318 . . . . . . . 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 4145 . . . . . 6 (πœ‘ β†’ 𝑆 ∈ V)
12 ressex 12528 . . . . . 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 12497 . . . . 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 1238 . . . 4 (πœ‘ β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
16 mulrslid 12593 . . . . . 6 (.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
1716slotex 12492 . . . . 5 (π‘Š ∈ 𝑋 β†’ (.rβ€˜π‘Š) ∈ V)
181, 17syl 14 . . . 4 (πœ‘ β†’ (.rβ€˜π‘Š) ∈ V)
19 vscandxnscandx 12623 . . . . . 6 ( ·𝑠 β€˜ndx) β‰  (Scalarβ€˜ndx)
2019necomi 2432 . . . . 5 (Scalarβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
21 vscaslid 12624 . . . . . 6 ( ·𝑠 = Slot ( ·𝑠 β€˜ndx) ∧ ( ·𝑠 β€˜ndx) ∈ β„•)
2221simpri 113 . . . . 5 ( ·𝑠 β€˜ndx) ∈ β„•
232, 20, 22setsslnid 12517 . . . 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 12497 . . . . 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 1238 . . . 4 (πœ‘ β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
28 slotsdifipndx 12636 . . . . . 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 12632 . . . . . 6 (·𝑖 = Slot (Β·π‘–β€˜ndx) ∧ (Β·π‘–β€˜ndx) ∈ β„•)
3130simpri 113 . . . . 5 (Β·π‘–β€˜ndx) ∈ β„•
322, 29, 31setsslnid 12517 . . . 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 2210 . 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 12516 . . 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 13562 . . . . 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 2210 . . 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 5521 . 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 2220 1 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1353   ∈ wcel 2148   β‰  wne 2347  Vcvv 2739   βŠ† wss 3131  βŸ¨cop 3597   Fn wfn 5213  β€˜cfv 5218  (class class class)co 5878  β„•cn 8922  ndxcnx 12462   sSet csts 12463  Slot cslot 12464  Basecbs 12465   β†Ύs cress 12466  .rcmulr 12540  Scalarcsca 12542   ·𝑠 cvsca 12543  Β·π‘–cip 12544  subringAlg csra 13558
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473  df-mulr 12553  df-sca 12555  df-vsca 12556  df-ip 12557  df-sra 13560
This theorem is referenced by:  sralmod  13575
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