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Theorem sraex 13635
Description: Existence of a subring algebra. (Contributed by Jim Kingdon, 16-Apr-2025.)
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
sraex (πœ‘ β†’ 𝐴 ∈ V)
Proof of Theorem sraex
StepHypRef Expression
1 srapart.a . 2 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
2 srapart.ex . . . 4 (πœ‘ β†’ π‘Š ∈ 𝑋)
3 srapart.s . . . 4 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
4 sraval 13626 . . . 4 ((π‘Š ∈ 𝑋 ∧ 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)⟩))
52, 3, 4syl2anc 411 . . 3 (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)⟩))
6 scaslid 12626 . . . . . . . 8 (Scalar = Slot (Scalarβ€˜ndx) ∧ (Scalarβ€˜ndx) ∈ β„•)
76simpri 113 . . . . . . 7 (Scalarβ€˜ndx) ∈ β„•
87a1i 9 . . . . . 6 (πœ‘ β†’ (Scalarβ€˜ndx) ∈ β„•)
9 basfn 12534 . . . . . . . . 9 Base Fn V
102elexd 2762 . . . . . . . . 9 (πœ‘ β†’ π‘Š ∈ V)
11 funfvex 5544 . . . . . . . . . 10 ((Fun Base ∧ π‘Š ∈ dom Base) β†’ (Baseβ€˜π‘Š) ∈ V)
1211funfni 5328 . . . . . . . . 9 ((Base Fn V ∧ π‘Š ∈ V) β†’ (Baseβ€˜π‘Š) ∈ V)
139, 10, 12sylancr 414 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜π‘Š) ∈ V)
1413, 3ssexd 4155 . . . . . . 7 (πœ‘ β†’ 𝑆 ∈ V)
15 ressex 12539 . . . . . . 7 ((π‘Š ∈ 𝑋 ∧ 𝑆 ∈ V) β†’ (π‘Š β†Ύs 𝑆) ∈ V)
162, 14, 15syl2anc 411 . . . . . 6 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) ∈ V)
17 setsex 12508 . . . . . 6 ((π‘Š ∈ 𝑋 ∧ (Scalarβ€˜ndx) ∈ β„• ∧ (π‘Š β†Ύs 𝑆) ∈ V) β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
182, 8, 16, 17syl3anc 1248 . . . . 5 (πœ‘ β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
19 vscaslid 12636 . . . . . . 7 ( ·𝑠 = Slot ( ·𝑠 β€˜ndx) ∧ ( ·𝑠 β€˜ndx) ∈ β„•)
2019simpri 113 . . . . . 6 ( ·𝑠 β€˜ndx) ∈ β„•
2120a1i 9 . . . . 5 (πœ‘ β†’ ( ·𝑠 β€˜ndx) ∈ β„•)
22 mulrslid 12605 . . . . . . 7 (.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
2322slotex 12503 . . . . . 6 (π‘Š ∈ 𝑋 β†’ (.rβ€˜π‘Š) ∈ V)
242, 23syl 14 . . . . 5 (πœ‘ β†’ (.rβ€˜π‘Š) ∈ V)
25 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)
2618, 21, 24, 25syl3anc 1248 . . . 4 (πœ‘ β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
27 ipslid 12644 . . . . . 6 (·𝑖 = Slot (Β·π‘–β€˜ndx) ∧ (Β·π‘–β€˜ndx) ∈ β„•)
2827simpri 113 . . . . 5 (Β·π‘–β€˜ndx) ∈ β„•
2928a1i 9 . . . 4 (πœ‘ β†’ (Β·π‘–β€˜ndx) ∈ β„•)
30 setsex 12508 . . . 4 ((((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V ∧ (Β·π‘–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)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
3126, 29, 24, 30syl3anc 1248 . . 3 (πœ‘ β†’ (((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)⟩) ∈ V)
325, 31eqeltrd 2264 . 2 (πœ‘ β†’ ((subringAlg β€˜π‘Š)β€˜π‘†) ∈ V)
331, 32eqeltrd 2264 1 (πœ‘ β†’ 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1363   ∈ wcel 2158  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-1re 7919  ax-addrcl 7922
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-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-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-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:  sratopng  13636  sralmod0g  13640  rlmfn  13642  rlmvalg  13643
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