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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.) |
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 | permsetex 18498 | . . . . 5 ⊢ (𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} ∈ V) | |
3 | eqid 2821 | . . . . . 6 ⊢ ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) | |
4 | eqid 2821 | . . . . . 6 ⊢ (+g‘(EndoFMnd‘𝐴)) = (+g‘(EndoFMnd‘𝐴)) | |
5 | 3, 4 | ressplusg 16612 | . . . . 5 ⊢ ({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 | syl 17 | . . . 4 ⊢ (𝐴 ∈ V → (+g‘(EndoFMnd‘𝐴)) = (+g‘((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}))) |
7 | symgplusg.1 | . . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐴) | |
8 | eqid 2821 | . . . . . . 7 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} = {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴} | |
9 | 7, 8 | symgval 18497 | . . . . . 6 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) |
10 | 9 | eqcomi 2830 | . . . . 5 ⊢ ((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴}) = 𝐺 |
11 | 10 | fveq2i 6673 | . . . 4 ⊢ (+g‘((EndoFMnd‘𝐴) ↾s {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐴})) = (+g‘𝐺) |
12 | 6, 11 | syl6eq 2872 | . . 3 ⊢ (𝐴 ∈ V → (+g‘(EndoFMnd‘𝐴)) = (+g‘𝐺)) |
13 | fvprc 6663 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
14 | fvprc 6663 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (SymGrp‘𝐴) = ∅) | |
15 | 7, 14 | syl5req 2869 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ∅ = 𝐺) |
16 | 13, 15 | eqtrd 2856 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = 𝐺) |
17 | 16 | fveq2d 6674 | . . 3 ⊢ (¬ 𝐴 ∈ V → (+g‘(EndoFMnd‘𝐴)) = (+g‘𝐺)) |
18 | 12, 17 | pm2.61i 184 | . 2 ⊢ (+g‘(EndoFMnd‘𝐴)) = (+g‘𝐺) |
19 | eqid 2821 | . . 3 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴) | |
20 | symgplusg.2 | . . . 4 ⊢ 𝐵 = (𝐴 ↑m 𝐴) | |
21 | eqid 2821 | . . . . 5 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴)) | |
22 | 19, 21 | efmndbas 18036 | . . . 4 ⊢ (Base‘(EndoFMnd‘𝐴)) = (𝐴 ↑m 𝐴) |
23 | 20, 22 | eqtr4i 2847 | . . 3 ⊢ 𝐵 = (Base‘(EndoFMnd‘𝐴)) |
24 | 19, 23, 4 | efmndplusg 18045 | . 2 ⊢ (+g‘(EndoFMnd‘𝐴)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
25 | 1, 18, 24 | 3eqtr2i 2850 | 1 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3494 ∅c0 4291 ∘ ccom 5559 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ↑m cmap 8406 Basecbs 16483 ↾s cress 16484 +gcplusg 16565 EndoFMndcefmnd 18033 SymGrpcsymg 18495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-efmnd 18034 df-symg 18496 |
This theorem is referenced by: symgov 18512 pgrpsubgsymg 18537 |
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