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

Theorem efmnd 18822
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 3490 . . 3 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ V)
3 ovexd 7455 . . . . 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 7438 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) = (𝐴 ↑m 𝐴))
7 efmnd.2 . . . . . . . . 9 𝐡 = (𝐴 ↑m 𝐴)
86, 7eqtr4di 2786 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (π‘Ž ↑m π‘Ž) = 𝐡)
94, 8sylan9eqr 2790 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ 𝑏 = 𝐡)
109opeq2d 4881 . . . . . 6 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐡⟩)
11 eqidd 2729 . . . . . . . . 9 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
129, 9, 11mpoeq123dv 7495 . . . . . . . 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 2786 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔)) = + )
1514opeq2d 4881 . . . . . 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 4617 . . . . . . . . . . . 12 (π‘Ž = 𝐴 β†’ 𝒫 π‘Ž = 𝒫 𝐴)
1817sneqd 4641 . . . . . . . . . . 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 5709 . . . . . . . . 9 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (π‘Ž Γ— {𝒫 π‘Ž}) = (𝐴 Γ— {𝒫 𝐴}))
2120fveq2d 6901 . . . . . . . 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 2786 . . . . . . 7 ((π‘Ž = 𝐴 ∧ 𝑏 = (π‘Ž ↑m π‘Ž)) β†’ (∏tβ€˜(π‘Ž Γ— {𝒫 π‘Ž})) = 𝐽)
2423opeq2d 4881 . . . . . 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 4753 . . . . 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 3930 . . . 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 18821 . . . 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 7749 . . . 4 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(+gβ€˜ndx), + ⟩, ⟨(TopSetβ€˜ndx), 𝐽⟩} ∈ V
2926, 27, 28fvmpt 7005 . . 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 2780 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 1534   ∈ wcel 2099  Vcvv 3471  β¦‹csb 3892  π’« cpw 4603  {csn 4629  {ctp 4633  βŸ¨cop 4635   Γ— cxp 5676   ∘ ccom 5682  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422   ↑m cmap 8845  ndxcnx 17162  Basecbs 17180  +gcplusg 17233  TopSetcts 17239  βˆtcpt 17420  EndoFMndcefmnd 18820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-efmnd 18821
This theorem is referenced by:  efmndbas  18823  efmndtset  18831  efmndplusg  18832  symgvalstruct  19351  symgvalstructOLD  19352
  Copyright terms: Public domain W3C validator