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Mirrors > Home > MPE Home > Th. List > symgtset | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
symggrp.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
Ref | Expression |
---|---|
symgtset | ⊢ (𝐴 ∈ 𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symggrp.1 | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | eqid 2610 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | symgbas 17623 | . . . 4 ⊢ (Base‘𝐺) = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
4 | eqid 2610 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | 1, 2, 4 | symgplusg 17632 | . . . 4 ⊢ (+g‘𝐺) = (𝑓 ∈ (Base‘𝐺), 𝑔 ∈ (Base‘𝐺) ↦ (𝑓 ∘ 𝑔)) |
6 | eqid 2610 | . . . 4 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴})) | |
7 | 1, 3, 5, 6 | symgval 17622 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {〈(Base‘ndx), (Base‘𝐺)〉, 〈(+g‘ndx), (+g‘𝐺)〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
8 | 7 | fveq2d 6107 | . 2 ⊢ (𝐴 ∈ 𝑉 → (TopSet‘𝐺) = (TopSet‘{〈(Base‘ndx), (Base‘𝐺)〉, 〈(+g‘ndx), (+g‘𝐺)〉, 〈(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), (+g‘𝐺)〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} = {〈(Base‘ndx), (Base‘𝐺)〉, 〈(+g‘ndx), (+g‘𝐺)〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} | |
11 | 10 | topgrptset 15868 | . . 3 ⊢ ((∏t‘(𝐴 × {𝒫 𝐴})) ∈ V → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘{〈(Base‘ndx), (Base‘𝐺)〉, 〈(+g‘ndx), (+g‘𝐺)〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
12 | 9, 11 | ax-mp 5 | . 2 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘{〈(Base‘ndx), (Base‘𝐺)〉, 〈(+g‘ndx), (+g‘𝐺)〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
13 | 8, 12 | syl6reqr 2663 | 1 ⊢ (𝐴 ∈ 𝑉 → (∏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 |
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