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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.) (Proof shortened by AV, 19-Feb-2024.) (Revised by AV, 29-Mar-2024.) (Proof shortened by AV, 14-Aug-2024.) |
Ref | Expression |
---|---|
symgplusg.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgplusg.2 | ⊢ 𝐵 = (𝐴 ↑m 𝐴) |
symgplusg.3 | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
symgplusg | ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgplusg.3 | . 2 ⊢ + = (+g‘𝐺) | |
2 | f1osetex 8529 | . . . 4 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V | |
3 | eqid 2734 | . . . . 5 ⊢ ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) | |
4 | eqid 2734 | . . . . 5 ⊢ (+g‘(EndoFMnd‘𝐴)) = (+g‘(EndoFMnd‘𝐴)) | |
5 | 3, 4 | ressplusg 16814 | . . . 4 ⊢ ({𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V → (+g‘(EndoFMnd‘𝐴)) = (+g‘((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}))) |
6 | 2, 5 | ax-mp 5 | . . 3 ⊢ (+g‘(EndoFMnd‘𝐴)) = (+g‘((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴})) |
7 | symgplusg.1 | . . . . . 6 ⊢ 𝐺 = (SymGrp‘𝐴) | |
8 | eqid 2734 | . . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} | |
9 | 7, 8 | symgval 18733 | . . . . 5 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) |
10 | 9 | eqcomi 2743 | . . . 4 ⊢ ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = 𝐺 |
11 | 10 | fveq2i 6709 | . . 3 ⊢ (+g‘((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴})) = (+g‘𝐺) |
12 | 6, 11 | eqtri 2762 | . 2 ⊢ (+g‘(EndoFMnd‘𝐴)) = (+g‘𝐺) |
13 | eqid 2734 | . . 3 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴) | |
14 | symgplusg.2 | . . . 4 ⊢ 𝐵 = (𝐴 ↑m 𝐴) | |
15 | eqid 2734 | . . . . 5 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴)) | |
16 | 13, 15 | efmndbas 18270 | . . . 4 ⊢ (Base‘(EndoFMnd‘𝐴)) = (𝐴 ↑m 𝐴) |
17 | 14, 16 | eqtr4i 2765 | . . 3 ⊢ 𝐵 = (Base‘(EndoFMnd‘𝐴)) |
18 | 13, 17, 4 | efmndplusg 18279 | . 2 ⊢ (+g‘(EndoFMnd‘𝐴)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
19 | 1, 12, 18 | 3eqtr2i 2768 | 1 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 {cab 2712 Vcvv 3401 ∘ ccom 5544 –1-1-onto→wf1o 6368 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 ↑m cmap 8497 Basecbs 16684 ↾s cress 16685 +gcplusg 16767 EndoFMndcefmnd 18267 SymGrpcsymg 18731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-tset 16786 df-efmnd 18268 df-symg 18732 |
This theorem is referenced by: symgov 18748 pgrpsubgsymg 18773 |
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