Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > symgsubg | Structured version Visualization version GIF version |
Description: The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
Ref | Expression |
---|---|
symgsubg.g | ⊢ 𝐺 = (SymGrp‘𝐴) |
symgsubg.b | ⊢ 𝐵 = (Base‘𝐺) |
symgsubg.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
symgsubg | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 ∘ ◡𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symgsubg.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2820 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2820 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | symgsubg.m | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 18145 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
6 | symgsubg.g | . . . . 5 ⊢ 𝐺 = (SymGrp‘𝐴) | |
7 | 6, 1, 3 | symginv 18526 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → ((invg‘𝐺)‘𝑌) = ◡𝑌) |
8 | 7 | adantl 484 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ◡𝑌) |
9 | 8 | oveq2d 7169 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋(+g‘𝐺)◡𝑌)) |
10 | 6, 1 | elbasfv 16540 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝐴 ∈ V) |
11 | 6 | symggrp 18524 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝐺 ∈ Grp) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ Grp) |
13 | 1, 3 | grpinvcl 18147 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
14 | 12, 13 | sylan 582 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
15 | 8, 14 | eqeltrrd 2913 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ◡𝑌 ∈ 𝐵) |
16 | 6, 1, 2 | symgov 18508 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ ◡𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)◡𝑌) = (𝑋 ∘ ◡𝑌)) |
17 | 15, 16 | syldan 593 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)◡𝑌) = (𝑋 ∘ ◡𝑌)) |
18 | 5, 9, 17 | 3eqtrd 2859 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 ∘ ◡𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3493 ◡ccnv 5551 ∘ ccom 5556 ‘cfv 6352 (class class class)co 7153 Basecbs 16479 +gcplusg 16561 Grpcgrp 18099 invgcminusg 18100 -gcsg 18101 SymGrpcsymg 18491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-oadd 8103 df-er 8286 df-map 8405 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-uz 12242 df-fz 12891 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-tset 16580 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-submnd 17953 df-efmnd 18030 df-grp 18102 df-minusg 18103 df-sbg 18104 df-symg 18492 |
This theorem is referenced by: cycpmconjs 30819 cyc3conja 30820 |
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