MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efmnd Structured version   Visualization version   GIF version

Theorem efmnd 18791
Description: The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
Hypotheses
Ref Expression
efmnd.1 𝐺 = (EndoFMndβ€˜π΄)
efmnd.2 𝐡 = (𝐴 ↑m 𝐴)
efmnd.3 + = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑓 ∘ 𝑔))
efmnd.4 𝐽 = (∏tβ€˜(𝐴 Γ— {𝒫 𝐴}))
Assertion
Ref Expression
efmnd (𝐴 ∈ 𝑉 β†’ 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
Distinct variable group:   𝑓,𝑔,𝐴
Allowed substitution hints:   𝐡(𝑓,𝑔)   + (𝑓,𝑔)   𝐺(𝑓,𝑔)   𝐽(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem efmnd
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efmnd.1 . 2 𝐺 = (EndoFMndβ€˜π΄)
2 elex 3485 . . 3 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ V)
3 ovexd 7437 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) ∈ V)
4 id 22 . . . . . . . 8 (𝑏 = (π‘Ž ↑m π‘Ž) β†’ 𝑏 = (π‘Ž ↑m π‘Ž))
5 id 22 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ π‘Ž = 𝐴)
65, 5oveq12d 7420 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) = (𝐴 ↑m 𝐴))
7 efmnd.2 . . . . . . . . 9 𝐡 = (𝐴 ↑m 𝐴)
86, 7eqtr4di 2782 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) = 𝐡)
94, 8sylan9eqr 2786 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ 𝑏 = 𝐡)
109opeq2d 4873 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐡⟩)
11 eqidd 2725 . . . . . . . . 9 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
129, 9, 11mpoeq123dv 7477 . . . . . . . 8 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑓 ∘ 𝑔)))
13 efmnd.3 . . . . . . . 8 + = (𝑓 ∈ 𝐡, 𝑔 ∈ 𝐡 ↦ (𝑓 ∘ 𝑔))
1412, 13eqtr4di 2782 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
1514opeq2d 4873 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+gβ€˜ndx), + ⟩)
16 simpl 482 . . . . . . . . . 10 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ π‘Ž = 𝐴)
17 pweq 4609 . . . . . . . . . . . 12 (π‘Ž = 𝐴 β†’ 𝒫 π‘Ž = 𝒫 𝐴)
1817sneqd 4633 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ {𝒫 π‘Ž} = {𝒫 𝐴})
1918adantr 480 . . . . . . . . . 10 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ {𝒫 π‘Ž} = {𝒫 𝐴})
2016, 19xpeq12d 5698 . . . . . . . . 9 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (π‘Ž Γ— {𝒫 π‘Ž}) = (𝐴 Γ— {𝒫 𝐴}))
2120fveq2d 6886 . . . . . . . 8 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž})) = (∏tβ€˜(𝐴 Γ— {𝒫 𝐴})))
22 efmnd.4 . . . . . . . 8 𝐽 = (∏tβ€˜(𝐴 Γ— {𝒫 𝐴}))
2321, 22eqtr4di 2782 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž})) = 𝐽)
2423opeq2d 4873 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩ = ⟨(TopSetβ€˜ndx), 𝐽⟩)
2510, 15, 24tpeq123d 4745 . . . . 5 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
263, 25csbied 3924 . . . 4 (π‘Ž = 𝐴 β†’ ⦋(π‘Ž ↑m π‘Ž) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
27 df-efmnd 18790 . . . 4 EndoFMnd = (π‘Ž ∈ V ↦ ⦋(π‘Ž ↑m π‘Ž) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(TopSetβ€˜ndx), (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž}))⟩})
28 tpex 7728 . . . 4 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩} ∈ V
2926, 27, 28fvmpt 6989 . . 3 (𝐴 ∈ V β†’ (EndoFMndβ€˜π΄) = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
302, 29syl 17 . 2 (𝐴 ∈ 𝑉 β†’ (EndoFMndβ€˜π΄) = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
311, 30eqtrid 2776 1 (𝐴 ∈ 𝑉 β†’ 𝐺 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  β¦‹csb 3886  π’« cpw 4595  {csn 4621  {ctp 4625  βŸ¨cop 4627   Γ— cxp 5665   ∘ ccom 5671  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404   ↑m cmap 8817  ndxcnx 17131  Basecbs 17149  +gcplusg 17202  TopSetcts 17208  βˆtcpt 17389  EndoFMndcefmnd 18789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-efmnd 18790
This theorem is referenced by:  efmndbas  18792  efmndtset  18800  efmndplusg  18801  symgvalstruct  19312  symgvalstructOLD  19313
  Copyright terms: Public domain W3C validator