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| Mirrors > Home > MPE Home > Th. List > grpsubval | Structured version Visualization version GIF version | ||
| Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| grpsubval.b | ⊢ 𝐵 = (Base‘𝐺) | 
| grpsubval.p | ⊢ + = (+g‘𝐺) | 
| grpsubval.i | ⊢ 𝐼 = (invg‘𝐺) | 
| grpsubval.m | ⊢ − = (-g‘𝐺) | 
| Ref | Expression | 
|---|---|
| grpsubval | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | oveq1 7439 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑦))) | |
| 2 | fveq2 6905 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
| 3 | 2 | oveq2d 7448 | . 2 ⊢ (𝑦 = 𝑌 → (𝑋 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑌))) | 
| 4 | grpsubval.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 5 | grpsubval.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 6 | grpsubval.i | . . 3 ⊢ 𝐼 = (invg‘𝐺) | |
| 7 | grpsubval.m | . . 3 ⊢ − = (-g‘𝐺) | |
| 8 | 4, 5, 6, 7 | grpsubfval 19002 | . 2 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) | 
| 9 | ovex 7465 | . 2 ⊢ (𝑋 + (𝐼‘𝑌)) ∈ V | |
| 10 | 1, 3, 8, 9 | ovmpo 7594 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 invgcminusg 18953 -gcsg 18954 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-sbg 18957 | 
| This theorem is referenced by: grpsubinv 19031 grpsubrcan 19040 grpinvsub 19041 grpinvval2 19042 grpsubid 19043 grpsubid1 19044 grpsubeq0 19045 grpsubadd0sub 19046 grpsubadd 19047 grpsubsub 19048 grpaddsubass 19049 grpnpcan 19051 pwssub 19073 mulgsubdir 19133 subgsubcl 19156 subgsub 19157 issubg4 19164 qussub 19210 ghmsub 19243 sylow2blem1 19639 lsmelvalm 19670 ablsub2inv 19827 ablsub4 19829 ablsubsub4 19837 mulgsubdi 19848 eqgabl 19853 gsumsub 19967 dprdfsub 20042 rngsubdi 20169 rngsubdir 20170 abvsubtri 20829 lmodvsubval2 20916 lmodsubdir 20919 lspsntrim 21098 cnfldsub 21411 m2detleiblem7 22634 chpscmatgsumbin 22851 tgpconncomp 24122 tsmssub 24158 tsmsxplem1 24162 isngp4 24626 ngpsubcan 24628 ngptgp 24650 tngngp3 24678 clmpm1dir 25137 cphipval 25278 deg1suble 26147 deg1sub 26148 dchr2sum 27318 ogrpsub 33094 symgsubg 33108 cycpmconjv 33163 archiabllem2c 33203 linds2eq 33410 ressply1sub 33596 r1padd1 33629 ply1divalg3 35648 lflsub 39069 ldualvsubval 39159 lcdvsubval 41621 baerlem3lem1 41710 baerlem5alem1 41711 baerlem5amN 41719 baerlem5bmN 41720 baerlem5abmN 41721 hdmapsub 41850 nelsubgsubcld 42513 | 
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