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

Theorem symgtset 17642
 Description: The topology of the symmetric group on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.)
Hypothesis
Ref Expression
symggrp.1 𝐺 = (SymGrp‘𝐴)
Assertion
Ref Expression
symgtset (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))

Proof of Theorem symgtset
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symggrp.1 . . . 4 𝐺 = (SymGrp‘𝐴)
2 eqid 2610 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
31, 2symgbas 17623 . . . 4 (Base‘𝐺) = {𝑥𝑥:𝐴1-1-onto𝐴}
4 eqid 2610 . . . . 5 (+g𝐺) = (+g𝐺)
51, 2, 4symgplusg 17632 . . . 4 (+g𝐺) = (𝑓 ∈ (Base‘𝐺), 𝑔 ∈ (Base‘𝐺) ↦ (𝑓𝑔))
6 eqid 2610 . . . 4 (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴}))
71, 3, 5, 6symgval 17622 . . 3 (𝐴𝑉𝐺 = {⟨(Base‘ndx), (Base‘𝐺)⟩, ⟨(+g‘ndx), (+g𝐺)⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩})
87fveq2d 6107 . 2 (𝐴𝑉 → (TopSet‘𝐺) = (TopSet‘{⟨(Base‘ndx), (Base‘𝐺)⟩, ⟨(+g‘ndx), (+g𝐺)⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}))
9 fvex 6113 . . 3 (∏t‘(𝐴 × {𝒫 𝐴})) ∈ V
10 eqid 2610 . . . 4 {⟨(Base‘ndx), (Base‘𝐺)⟩, ⟨(+g‘ndx), (+g𝐺)⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩} = {⟨(Base‘ndx), (Base‘𝐺)⟩, ⟨(+g‘ndx), (+g𝐺)⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}
1110topgrptset 15868 . . 3 ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ V → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘{⟨(Base‘ndx), (Base‘𝐺)⟩, ⟨(+g‘ndx), (+g𝐺)⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩}))
129, 11ax-mp 5 . 2 (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘{⟨(Base‘ndx), (Base‘𝐺)⟩, ⟨(+g‘ndx), (+g𝐺)⟩, ⟨(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))⟩})
138, 12syl6reqr 2663 1 (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108  {csn 4125  {ctp 4129  ⟨cop 4131   × cxp 5036  ‘cfv 5804  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  TopSetcts 15774  ∏tcpt 15922  SymGrpcsymg 17620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-tset 15787  df-symg 17621 This theorem is referenced by:  symgtopn  17648
 Copyright terms: Public domain W3C validator