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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 6657 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑦))) | |
2 | fveq2 6191 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
3 | 2 | oveq2d 6666 | . 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 17464 | . 2 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) |
9 | ovex 6678 | . 2 ⊢ (𝑋 + (𝐼‘𝑌)) ∈ V | |
10 | 1, 3, 8, 9 | ovmpt2 6796 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 invgcminusg 17423 -gcsg 17424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-sbg 17427 |
This theorem is referenced by: grpsubinv 17488 grpsubrcan 17496 grpinvsub 17497 grpinvval2 17498 grpsubid 17499 grpsubid1 17500 grpsubeq0 17501 grpsubadd0sub 17502 grpsubadd 17503 grpsubsub 17504 grpaddsubass 17505 grpnpcan 17507 pwssub 17529 mulgsubdir 17582 subgsubcl 17605 subgsub 17606 issubg4 17613 qussub 17654 ghmsub 17668 sylow2blem1 18035 lsmelvalm 18066 ablsub2inv 18216 ablsub4 18218 ablsubsub4 18224 mulgsubdi 18235 eqgabl 18240 gsumsub 18348 dprdfsub 18420 ringsubdi 18599 rngsubdir 18600 abvsubtri 18835 lmodvsubval2 18918 lmodsubdir 18921 lspsntrim 19098 cnfldsub 19774 m2detleiblem7 20433 chpscmatgsumbin 20649 tgpconncomp 21916 tsmssub 21952 tsmsxplem1 21956 isngp4 22416 ngpsubcan 22418 ngptgp 22440 tngngp3 22460 clmpm1dir 22903 cphipval 23042 deg1suble 23867 deg1sub 23868 dchr2sum 24998 ogrpsub 29717 archiabllem2c 29749 lflsub 34354 ldualvsubval 34444 lcdvsubval 36907 baerlem3lem1 36996 baerlem5alem1 36997 baerlem5amN 37005 baerlem5bmN 37006 baerlem5abmN 37007 hdmapsub 37139 |
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