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Mirrors > Home > MPE Home > Th. List > symgplusg | Structured version Visualization version GIF version |
Description: The group operation of a symmetric group is the function composition. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) |
Ref | Expression |
---|---|
symgplusg.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgplusg.2 | ⊢ 𝐵 = (Base‘𝐺) |
symgplusg.3 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
symgplusg | ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgplusg.1 | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝐴) | |
2 | symgplusg.2 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
3 | 1, 2 | symgbas 17623 | . . . . 5 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} |
4 | eqid 2610 | . . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) | |
5 | eqid 2610 | . . . . 5 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴})) | |
6 | 1, 3, 4, 5 | symgval 17622 | . . . 4 ⊢ (𝐴 ∈ V → 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
7 | 6 | fveq2d 6107 | . . 3 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
8 | symgplusg.3 | . . 3 ⊢ + = (+g‘𝐺) | |
9 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
10 | 2, 9 | eqeltri 2684 | . . . . 5 ⊢ 𝐵 ∈ V |
11 | 10, 10 | mpt2ex 7136 | . . . 4 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) ∈ V |
12 | eqid 2610 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} | |
13 | 12 | topgrpplusg 15867 | . . . 4 ⊢ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) ∈ V → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = (+g‘{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
14 | 11, 13 | ax-mp 5 | . . 3 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = (+g‘{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
15 | 7, 8, 14 | 3eqtr4g 2669 | . 2 ⊢ (𝐴 ∈ V → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))) |
16 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅) | |
17 | 1, 16 | syl5eq 2656 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐺 = ∅) |
18 | 17 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (+g‘𝐺) = (+g‘∅)) |
19 | plusgid 15804 | . . . . 5 ⊢ +g = Slot (+g‘ndx) | |
20 | 19 | str0 15739 | . . . 4 ⊢ ∅ = (+g‘∅) |
21 | 18, 8, 20 | 3eqtr4g 2669 | . . 3 ⊢ (¬ 𝐴 ∈ V → + = ∅) |
22 | vex 3176 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
23 | vex 3176 | . . . . . . 7 ⊢ 𝑔 ∈ V | |
24 | 22, 23 | coex 7011 | . . . . . 6 ⊢ (𝑓 ∘ 𝑔) ∈ V |
25 | 4, 24 | fnmpt2i 7128 | . . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) Fn (𝐵 × 𝐵) |
26 | 17 | fveq2d 6107 | . . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V → (Base‘𝐺) = (Base‘∅)) |
27 | base0 15740 | . . . . . . . . 9 ⊢ ∅ = (Base‘∅) | |
28 | 26, 2, 27 | 3eqtr4g 2669 | . . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → 𝐵 = ∅) |
29 | 28 | xpeq2d 5063 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐵 × 𝐵) = (𝐵 × ∅)) |
30 | xp0 5471 | . . . . . . 7 ⊢ (𝐵 × ∅) = ∅ | |
31 | 29, 30 | syl6eq 2660 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐵 × 𝐵) = ∅) |
32 | 31 | fneq2d 5896 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) Fn (𝐵 × 𝐵) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) Fn ∅)) |
33 | 25, 32 | mpbii 222 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) Fn ∅) |
34 | fn0 5924 | . . . 4 ⊢ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) Fn ∅ ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = ∅) | |
35 | 33, 34 | sylib 207 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = ∅) |
36 | 21, 35 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐴 ∈ V → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))) |
37 | 15, 36 | pm2.61i 175 | 1 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {ctp 4129 〈cop 4131 × cxp 5036 ∘ ccom 5042 Fn wfn 5799 ‘cfv 5804 ↦ cmpt2 6551 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: symgov 17633 symgtset 17642 pgrpsubgsymg 17651 symgtgp 21715 |
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