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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 2731 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | eqid 2731 | . . 3 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | symgsubg.m | . . 3 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 18845 | . 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 19234 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → ((invg‘𝐺)‘𝑌) = ◡𝑌) |
8 | 7 | adantl 482 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ◡𝑌) |
9 | 8 | oveq2d 7409 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋(+g‘𝐺)◡𝑌)) |
10 | 6, 1 | elbasfv 17132 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝐴 ∈ V) |
11 | 6 | symggrp 19232 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝐺 ∈ Grp) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ Grp) |
13 | 1, 3 | grpinvcl 18847 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
14 | 12, 13 | sylan 580 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
15 | 8, 14 | eqeltrrd 2833 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ◡𝑌 ∈ 𝐵) |
16 | 6, 1, 2 | symgov 19215 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ ◡𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)◡𝑌) = (𝑋 ∘ ◡𝑌)) |
17 | 15, 16 | syldan 591 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)◡𝑌) = (𝑋 ∘ ◡𝑌)) |
18 | 5, 9, 17 | 3eqtrd 2775 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 ∘ ◡𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ◡ccnv 5668 ∘ ccom 5673 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 Grpcgrp 18794 invgcminusg 18795 -gcsg 18796 SymGrpcsymg 19198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-tset 17198 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-efmnd 18725 df-grp 18797 df-minusg 18798 df-sbg 18799 df-symg 19199 |
This theorem is referenced by: cycpmconjs 32186 cyc3conja 32187 |
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