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

Theorem symgval 17727
Description: The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
symgval.1 𝐺 = (SymGrp‘𝐴)
symgval.2 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
symgval.3 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
symgval.4 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
Assertion
Ref Expression
symgval (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Distinct variable group:   𝑓,𝑔,𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑓,𝑔)   + (𝑥,𝑓,𝑔)   𝐺(𝑥,𝑓,𝑔)   𝐽(𝑥,𝑓,𝑔)   𝑉(𝑥,𝑓,𝑔)

Proof of Theorem symgval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgval.1 . 2 𝐺 = (SymGrp‘𝐴)
2 elex 3201 . . 3 (𝐴𝑉𝐴 ∈ V)
3 ovex 6638 . . . . . . 7 (𝑎𝑚 𝑎) ∈ V
4 f1of 6099 . . . . . . . . 9 (𝑥:𝑎1-1-onto𝑎𝑥:𝑎𝑎)
5 vex 3192 . . . . . . . . . 10 𝑎 ∈ V
65, 5elmap 7837 . . . . . . . . 9 (𝑥 ∈ (𝑎𝑚 𝑎) ↔ 𝑥:𝑎𝑎)
74, 6sylibr 224 . . . . . . . 8 (𝑥:𝑎1-1-onto𝑎𝑥 ∈ (𝑎𝑚 𝑎))
87abssi 3661 . . . . . . 7 {𝑥𝑥:𝑎1-1-onto𝑎} ⊆ (𝑎𝑚 𝑎)
93, 8ssexi 4768 . . . . . 6 {𝑥𝑥:𝑎1-1-onto𝑎} ∈ V
109a1i 11 . . . . 5 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} ∈ V)
11 id 22 . . . . . . . 8 (𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎} → 𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎})
12 f1oeq23 6092 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1312anidms 676 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑥:𝑎1-1-onto𝑎𝑥:𝐴1-1-onto𝐴))
1413abbidv 2738 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = {𝑥𝑥:𝐴1-1-onto𝐴})
15 symgval.2 . . . . . . . . 9 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
1614, 15syl6eqr 2673 . . . . . . . 8 (𝑎 = 𝐴 → {𝑥𝑥:𝑎1-1-onto𝑎} = 𝐵)
1711, 16sylan9eqr 2677 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑏 = 𝐵)
1817opeq2d 4382 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
19 eqidd 2622 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑔) = (𝑓𝑔))
2017, 17, 19mpt2eq123dv 6677 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔)))
21 symgval.3 . . . . . . . 8 + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
2220, 21syl6eqr 2673 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔)) = + )
2322opeq2d 4382 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩ = ⟨(+g‘ndx), + ⟩)
24 simpl 473 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝑎 = 𝐴)
2524pweqd 4140 . . . . . . . . . . 11 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → 𝒫 𝑎 = 𝒫 𝐴)
2625sneqd 4165 . . . . . . . . . 10 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {𝒫 𝑎} = {𝒫 𝐴})
2724, 26xpeq12d 5105 . . . . . . . . 9 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (𝑎 × {𝒫 𝑎}) = (𝐴 × {𝒫 𝐴}))
2827fveq2d 6157 . . . . . . . 8 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = (∏t‘(𝐴 × {𝒫 𝐴})))
29 symgval.4 . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
3028, 29syl6eqr 2673 . . . . . . 7 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → (∏t‘(𝑎 × {𝒫 𝑎})) = 𝐽)
3130opeq2d 4382 . . . . . 6 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩ = ⟨(TopSet‘ndx), 𝐽⟩)
3218, 23, 31tpeq123d 4258 . . . . 5 ((𝑎 = 𝐴𝑏 = {𝑥𝑥:𝑎1-1-onto𝑎}) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3310, 32csbied 3545 . . . 4 (𝑎 = 𝐴{𝑥𝑥:𝑎1-1-onto𝑎} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
34 df-symg 17726 . . . 4 SymGrp = (𝑎 ∈ V ↦ {𝑥𝑥:𝑎1-1-onto𝑎} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑎 × {𝒫 𝑎}))⟩})
35 tpex 6917 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∈ V
3633, 34, 35fvmpt 6244 . . 3 (𝐴 ∈ V → (SymGrp‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
372, 36syl 17 . 2 (𝐴𝑉 → (SymGrp‘𝐴) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
381, 37syl5eq 2667 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  Vcvv 3189  csb 3518  𝒫 cpw 4135  {csn 4153  {ctp 4157  cop 4159   × cxp 5077  ccom 5083  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  𝑚 cmap 7809  ndxcnx 15785  Basecbs 15788  +gcplusg 15869  TopSetcts 15875  tcpt 16027  SymGrpcsymg 17725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-symg 17726
This theorem is referenced by:  symgbas  17728  symgplusg  17737  symgtset  17747
  Copyright terms: Public domain W3C validator