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Theorem sralemg 13627
Description: Lemma for srabaseg 13628 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-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)
sralemg.1 (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)
sralem.2 (Scalarβ€˜ndx) β‰  (πΈβ€˜ndx)
sralem.3 ( ·𝑠 β€˜ndx) β‰  (πΈβ€˜ndx)
sralem.4 (Β·π‘–β€˜ndx) β‰  (πΈβ€˜ndx)
Assertion
Ref Expression
sralemg (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
Proof of Theorem sralemg
StepHypRef Expression
1 srapart.ex . . . 4 (πœ‘ β†’ π‘Š ∈ 𝑋)
2 basfn 12534 . . . . . . 7 Base Fn V
31elexd 2762 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ V)
4 funfvex 5544 . . . . . . . 8 ((Fun Base ∧ π‘Š ∈ dom Base) β†’ (Baseβ€˜π‘Š) ∈ V)
54funfni 5328 . . . . . . 7 ((Base Fn V ∧ π‘Š ∈ V) β†’ (Baseβ€˜π‘Š) ∈ V)
62, 3, 5sylancr 414 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘Š) ∈ V)
7 srapart.s . . . . . 6 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
86, 7ssexd 4155 . . . . 5 (πœ‘ β†’ 𝑆 ∈ V)
9 ressex 12539 . . . . 5 ((π‘Š ∈ 𝑋 ∧ 𝑆 ∈ V) β†’ (π‘Š β†Ύs 𝑆) ∈ V)
101, 8, 9syl2anc 411 . . . 4 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) ∈ V)
11 sralemg.1 . . . . 5 (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)
12 sralem.2 . . . . . 6 (Scalarβ€˜ndx) β‰  (πΈβ€˜ndx)
1312necomi 2442 . . . . 5 (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)
14 scaslid 12626 . . . . . 6 (Scalar = Slot (Scalarβ€˜ndx) ∧ (Scalarβ€˜ndx) ∈ β„•)
1514simpri 113 . . . . 5 (Scalarβ€˜ndx) ∈ β„•
1611, 13, 15setsslnid 12528 . . . 4 ((π‘Š ∈ 𝑋 ∧ (π‘Š β†Ύs 𝑆) ∈ V) β†’ (πΈβ€˜π‘Š) = (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)))
171, 10, 16syl2anc 411 . . 3 (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)))
1815a1i 9 . . . . 5 (πœ‘ β†’ (Scalarβ€˜ndx) ∈ β„•)
19 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)
201, 18, 10, 19syl3anc 1248 . . . 4 (πœ‘ β†’ (π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V)
21 mulrslid 12605 . . . . . 6 (.r = Slot (.rβ€˜ndx) ∧ (.rβ€˜ndx) ∈ β„•)
2221slotex 12503 . . . . 5 (π‘Š ∈ 𝑋 β†’ (.rβ€˜π‘Š) ∈ V)
231, 22syl 14 . . . 4 (πœ‘ β†’ (.rβ€˜π‘Š) ∈ V)
24 sralem.3 . . . . . 6 ( ·𝑠 β€˜ndx) β‰  (πΈβ€˜ndx)
2524necomi 2442 . . . . 5 (πΈβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)
26 vscaslid 12636 . . . . . 6 ( ·𝑠 = Slot ( ·𝑠 β€˜ndx) ∧ ( ·𝑠 β€˜ndx) ∈ β„•)
2726simpri 113 . . . . 5 ( ·𝑠 β€˜ndx) ∈ β„•
2811, 25, 27setsslnid 12528 . . . 4 (((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) ∈ V ∧ (.rβ€˜π‘Š) ∈ V) β†’ (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)) = (πΈβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)))
2920, 23, 28syl2anc 411 . . 3 (πœ‘ β†’ (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩)) = (πΈβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)))
3027a1i 9 . . . . 5 (πœ‘ β†’ ( ·𝑠 β€˜ndx) ∈ β„•)
31 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)
3220, 30, 23, 31syl3anc 1248 . . . 4 (πœ‘ β†’ ((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩) ∈ V)
33 sralem.4 . . . . . 6 (Β·π‘–β€˜ndx) β‰  (πΈβ€˜ndx)
3433necomi 2442 . . . . 5 (πΈβ€˜ndx) β‰  (Β·π‘–β€˜ndx)
35 ipslid 12644 . . . . . 6 (·𝑖 = Slot (Β·π‘–β€˜ndx) ∧ (Β·π‘–β€˜ndx) ∈ β„•)
3635simpri 113 . . . . 5 (Β·π‘–β€˜ndx) ∈ β„•
3711, 34, 36setsslnid 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†’ (πΈβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)) = (πΈβ€˜(((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)⟩)))
3832, 23, 37syl2anc 411 . . 3 (πœ‘ β†’ (πΈβ€˜((π‘Š sSet ⟨(Scalarβ€˜ndx), (π‘Š β†Ύs 𝑆)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘Š)⟩)) = (πΈβ€˜(((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)⟩)))
3917, 29, 383eqtrd 2224 . 2 (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜(((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)⟩)))
40 srapart.a . . . 4 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
41 sraval 13626 . . . . 5 ((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)⟩))
421, 7, 41syl2anc 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)⟩))
4340, 42eqtrd 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)⟩))
4443fveq2d 5531 . 2 (πœ‘ β†’ (πΈβ€˜π΄) = (πΈβ€˜(((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)⟩)))
4539, 44eqtr4d 2223 1 (πœ‘ β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π΄))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   = 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-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-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-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:  srabaseg  13628  sraaddgg  13629  sramulrg  13630  sratsetg  13634  sradsg  13637
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