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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 18913 | . 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 19318 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → ((invg‘𝐺)‘𝑌) = ◡𝑌) |
8 | 7 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ◡𝑌) |
9 | 8 | oveq2d 7428 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋(+g‘𝐺)◡𝑌)) |
10 | 6, 1 | elbasfv 17157 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → 𝐴 ∈ V) |
11 | 6 | symggrp 19316 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝐺 ∈ Grp) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ Grp) |
13 | 1, 3 | grpinvcl 18915 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
14 | 12, 13 | sylan 579 | . . . 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 19299 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ ◡𝑌 ∈ 𝐵) → (𝑋(+g‘𝐺)◡𝑌) = (𝑋 ∘ ◡𝑌)) |
17 | 15, 16 | syldan 590 | . 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 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ◡ccnv 5675 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +gcplusg 17204 Grpcgrp 18861 invgcminusg 18862 -gcsg 18863 SymGrpcsymg 19282 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-tset 17223 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-efmnd 18792 df-grp 18864 df-minusg 18865 df-sbg 18866 df-symg 19283 |
This theorem is referenced by: cycpmconjs 32753 cyc3conja 32754 |
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