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Theorem sralemg 13684
Description: Lemma for srabaseg 13685 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 (𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))
srapart.s (𝜑𝑆 ⊆ (Base‘𝑊))
srapart.ex (𝜑𝑊𝑋)
sralemg.1 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
sralem.2 (Scalar‘ndx) ≠ (𝐸‘ndx)
sralem.3 ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
sralem.4 (·𝑖‘ndx) ≠ (𝐸‘ndx)
Assertion
Ref Expression
sralemg (𝜑 → (𝐸𝑊) = (𝐸𝐴))

Proof of Theorem sralemg
StepHypRef Expression
1 srapart.ex . . . 4 (𝜑𝑊𝑋)
2 basfn 12534 . . . . . . 7 Base Fn V
31elexd 2762 . . . . . . 7 (𝜑𝑊 ∈ V)
4 funfvex 5544 . . . . . . . 8 ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
54funfni 5328 . . . . . . 7 ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
62, 3, 5sylancr 414 . . . . . 6 (𝜑 → (Base‘𝑊) ∈ V)
7 srapart.s . . . . . 6 (𝜑𝑆 ⊆ (Base‘𝑊))
86, 7ssexd 4155 . . . . 5 (𝜑𝑆 ∈ V)
9 ressex 12539 . . . . 5 ((𝑊𝑋𝑆 ∈ V) → (𝑊s 𝑆) ∈ V)
101, 8, 9syl2anc 411 . . . 4 (𝜑 → (𝑊s 𝑆) ∈ V)
11 sralemg.1 . . . . 5 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
12 sralem.2 . . . . . 6 (Scalar‘ndx) ≠ (𝐸‘ndx)
1312necomi 2442 . . . . 5 (𝐸‘ndx) ≠ (Scalar‘ndx)
14 scaslid 12626 . . . . . 6 (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
1514simpri 113 . . . . 5 (Scalar‘ndx) ∈ ℕ
1611, 13, 15setsslnid 12528 . . . 4 ((𝑊𝑋 ∧ (𝑊s 𝑆) ∈ V) → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)))
171, 10, 16syl2anc 411 . . 3 (𝜑 → (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)))
1815a1i 9 . . . . 5 (𝜑 → (Scalar‘ndx) ∈ ℕ)
19 setsex 12508 . . . . 5 ((𝑊𝑋 ∧ (Scalar‘ndx) ∈ ℕ ∧ (𝑊s 𝑆) ∈ V) → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V)
201, 18, 10, 19syl3anc 1248 . . . 4 (𝜑 → (𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V)
21 mulrslid 12605 . . . . . 6 (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
2221slotex 12503 . . . . 5 (𝑊𝑋 → (.r𝑊) ∈ V)
231, 22syl 14 . . . 4 (𝜑 → (.r𝑊) ∈ V)
24 sralem.3 . . . . . 6 ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx)
2524necomi 2442 . . . . 5 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
26 vscaslid 12636 . . . . . 6 ( ·𝑠 = Slot ( ·𝑠 ‘ndx) ∧ ( ·𝑠 ‘ndx) ∈ ℕ)
2726simpri 113 . . . . 5 ( ·𝑠 ‘ndx) ∈ ℕ
2811, 25, 27setsslnid 12528 . . . 4 (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V ∧ (.r𝑊) ∈ V) → (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)))
2920, 23, 28syl2anc 411 . . 3 (𝜑 → (𝐸‘(𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩)) = (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)))
3027a1i 9 . . . . 5 (𝜑 → ( ·𝑠 ‘ndx) ∈ ℕ)
31 setsex 12508 . . . . 5 (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) ∈ V ∧ ( ·𝑠 ‘ndx) ∈ ℕ ∧ (.r𝑊) ∈ V) → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) ∈ V)
3220, 30, 23, 31syl3anc 1248 . . . 4 (𝜑 → ((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) ∈ V)
33 sralem.4 . . . . . 6 (·𝑖‘ndx) ≠ (𝐸‘ndx)
3433necomi 2442 . . . . 5 (𝐸‘ndx) ≠ (·𝑖‘ndx)
35 ipslid 12644 . . . . . 6 (·𝑖 = Slot (·𝑖‘ndx) ∧ (·𝑖‘ndx) ∈ ℕ)
3635simpri 113 . . . . 5 (·𝑖‘ndx) ∈ ℕ
3711, 34, 36setsslnid 12528 . . . 4 ((((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) ∈ V ∧ (.r𝑊) ∈ V) → (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
3832, 23, 37syl2anc 411 . . 3 (𝜑 → (𝐸‘((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩)) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
3917, 29, 383eqtrd 2224 . 2 (𝜑 → (𝐸𝑊) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
40 srapart.a . . . 4 (𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))
41 sraval 13683 . . . . 5 ((𝑊𝑋𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
421, 7, 41syl2anc 411 . . . 4 (𝜑 → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
4340, 42eqtrd 2220 . . 3 (𝜑𝐴 = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))
4443fveq2d 5531 . 2 (𝜑 → (𝐸𝐴) = (𝐸‘(((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩)))
4539, 44eqtr4d 2223 1 (𝜑 → (𝐸𝑊) = (𝐸𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  wne 2357  Vcvv 2749  wss 3141  cop 3607   Fn wfn 5223  cfv 5228  (class class class)co 5888  cn 8933  ndxcnx 12473   sSet csts 12474  Slot cslot 12475  Basecbs 12476  s cress 12477  .rcmulr 12552  Scalarcsca 12554   ·𝑠 cvsca 12555  ·𝑖cip 12556  subringAlg csra 13679
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 13681
This theorem is referenced by:  srabaseg  13685  sraaddgg  13686  sramulrg  13687  sratsetg  13691  sradsg  13694
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