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Theorem sraval 13626
Description: Lemma for srabaseg 13628 through sravscag 13632. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Assertion
Ref Expression
sraval ((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)⟩))
Proof of Theorem sraval
Dummy variables 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2760 . . . 4 (π‘Š ∈ 𝑉 β†’ π‘Š ∈ V)
21adantr 276 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ π‘Š ∈ V)
3 df-sra 13624 . . . 4 subringAlg = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩)))
4 fveq2 5527 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
54pweqd 3592 . . . . 5 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 (Baseβ€˜π‘Š))
6 id 19 . . . . . . . 8 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
7 oveq1 5895 . . . . . . . . 9 (𝑀 = π‘Š β†’ (𝑀 β†Ύs 𝑠) = (π‘Š β†Ύs 𝑠))
87opeq2d 3797 . . . . . . . 8 (𝑀 = π‘Š β†’ ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩ = ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩)
96, 8oveq12d 5906 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) = (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩))
10 fveq2 5527 . . . . . . . 8 (𝑀 = π‘Š β†’ (.rβ€˜π‘€) = (.rβ€˜π‘Š))
1110opeq2d 3797 . . . . . . 7 (𝑀 = π‘Š β†’ ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩ = ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)
129, 11oveq12d 5906 . . . . . 6 (𝑀 = π‘Š β†’ ((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) = ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩))
1310opeq2d 3797 . . . . . 6 (𝑀 = π‘Š β†’ ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩ = ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘Š)⟩)
1412, 13oveq12d 5906 . . . . 5 (𝑀 = π‘Š β†’ (((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜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)⟩))
155, 14mpteq12dv 4097 . . . 4 (𝑀 = π‘Š β†’ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (((𝑀 sSet ⟨(Scalarβ€˜ndx), (𝑀 β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘€)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘€)⟩)) = (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((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)⟩)))
16 elex 2760 . . . 4 (π‘Š ∈ V β†’ π‘Š ∈ V)
17 basfn 12534 . . . . . . 7 Base Fn V
18 funfvex 5544 . . . . . . . 8 ((Fun Base ∧ π‘Š ∈ dom Base) β†’ (Baseβ€˜π‘Š) ∈ V)
1918funfni 5328 . . . . . . 7 ((Base Fn V ∧ π‘Š ∈ V) β†’ (Baseβ€˜π‘Š) ∈ V)
2017, 19mpan 424 . . . . . 6 (π‘Š ∈ V β†’ (Baseβ€˜π‘Š) ∈ V)
2120pwexd 4193 . . . . 5 (π‘Š ∈ V β†’ 𝒫 (Baseβ€˜π‘Š) ∈ V)
2221mptexd 5756 . . . 4 (π‘Š ∈ V β†’ (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((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)
233, 15, 16, 22fvmptd3 5622 . . 3 (π‘Š ∈ V β†’ (subringAlg β€˜π‘Š) = (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((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)⟩)))
242, 23syl 14 . 2 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (subringAlg β€˜π‘Š) = (𝑠 ∈ 𝒫 (Baseβ€˜π‘Š) ↦ (((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)⟩)))
25 simpr 110 . . . . . . 7 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ 𝑠 = 𝑆)
2625oveq2d 5904 . . . . . 6 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ (π‘Š β†Ύs 𝑠) = (π‘Š β†Ύs 𝑆))
2726opeq2d 3797 . . . . 5 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩ = ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)
2827oveq2d 5904 . . . 4 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) = (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩))
2928oveq1d 5903 . . 3 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑠)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) = ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩))
3029oveq1d 5903 . 2 (((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) ∧ 𝑠 = 𝑆) 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)⟩) = (((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)⟩))
31 simpr 110 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
32 elpw2g 4168 . . . 4 ((Baseβ€˜π‘Š) ∈ V β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑆 βŠ† (Baseβ€˜π‘Š)))
332, 20, 323syl 17 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (𝑆 ∈ 𝒫 (Baseβ€˜π‘Š) ↔ 𝑆 βŠ† (Baseβ€˜π‘Š)))
3431, 33mpbird 167 . 2 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ 𝑆 ∈ 𝒫 (Baseβ€˜π‘Š))
35 simpl 109 . . . . 5 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ π‘Š ∈ 𝑉)
36 scaslid 12626 . . . . . . 7 (Scalar = Slot (Scalarβ€˜ndx) ∧ (Scalarβ€˜ndx) ∈ β„•)
3736simpri 113 . . . . . 6 (Scalarβ€˜ndx) ∈ β„•
3837a1i 9 . . . . 5 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (Scalarβ€˜ndx) ∈ β„•)
3934elexd 2762 . . . . . 6 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ 𝑆 ∈ V)
40 ressex 12539 . . . . . 6 ((π‘Š ∈ 𝑉 ∧ 𝑆 ∈ V) β†’ (π‘Š β†Ύs 𝑆) ∈ V)
4139, 40syldan 282 . . . . 5 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (π‘Š β†Ύs 𝑆) ∈ V)
42 setsex 12508 . . . . 5 ((π‘Š ∈ 𝑉 ∧ (Scalarβ€˜ndx) ∈ β„• ∧ (π‘Š β†Ύs 𝑆) ∈ V) β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
4335, 38, 41, 42syl3anc 1248 . . . 4 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
44 vscaslid 12636 . . . . . 6 ( ·𝑠 = Slot ( ·𝑠 β€˜ndx) ∧ ( ·𝑠 β€˜ndx) ∈ β„•)
4544simpri 113 . . . . 5 ( ·𝑠 β€˜ndx) ∈ β„•
4645a1i 9 . . . 4 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ ( ·𝑠 β€˜ndx) ∈ β„•)
47 mulrslid 12605 . . . . . 6 (.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
4847slotex 12503 . . . . 5 (π‘Š ∈ 𝑉 β†’ (.rβ€˜π‘Š) ∈ V)
4948adantr 276 . . . 4 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (.rβ€˜π‘Š) ∈ V)
50 setsex 12508 . . . 4 (((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)
5143, 46, 49, 50syl3anc 1248 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
52 ipslid 12644 . . . . 5 (·𝑖 = Slot (Β·π‘–β€˜ndx) ∧ (Β·π‘–β€˜ndx) ∈ β„•)
5352simpri 113 . . . 4 (Β·π‘–β€˜ndx) ∈ β„•
5453a1i 9 . . 3 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ (Β·π‘–β€˜ndx) ∈ β„•)
55 setsex 12508 . . 3 ((((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)
5651, 54, 49, 55syl3anc 1248 . 2 ((π‘Š ∈ 𝑉 ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) 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)
5724, 30, 34, 56fvmptd 5610 1 ((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)⟩))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363   ∈ wcel 2158  Vcvv 2749   βŠ† wss 3141  π’« cpw 3587  βŸ¨cop 3607   ↦ cmpt 4076   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:  sralemg  13627  srascag  13631  sravscag  13632  sraipg  13633  sraex  13635
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