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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 7415 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 + (𝐼‘𝑦)) = (𝑋 + (𝐼‘𝑦))) | |
2 | fveq2 6891 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
3 | 2 | oveq2d 7424 | . 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 18867 | . 2 ⊢ − = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + (𝐼‘𝑦))) |
9 | ovex 7441 | . 2 ⊢ (𝑋 + (𝐼‘𝑌)) ∈ V | |
10 | 1, 3, 8, 9 | ovmpo 7567 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + (𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +gcplusg 17196 invgcminusg 18819 -gcsg 18820 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-sbg 18823 |
This theorem is referenced by: grpsubinv 18895 grpsubrcan 18903 grpinvsub 18904 grpinvval2 18905 grpsubid 18906 grpsubid1 18907 grpsubeq0 18908 grpsubadd0sub 18909 grpsubadd 18910 grpsubsub 18911 grpaddsubass 18912 grpnpcan 18914 pwssub 18936 mulgsubdir 18993 subgsubcl 19016 subgsub 19017 issubg4 19024 qussub 19069 ghmsub 19099 sylow2blem1 19487 lsmelvalm 19518 ablsub2inv 19675 ablsub4 19677 ablsubsub4 19685 mulgsubdi 19696 eqgabl 19701 gsumsub 19815 dprdfsub 19890 ringsubdi 20118 ringsubdir 20119 abvsubtri 20442 lmodvsubval2 20526 lmodsubdir 20529 lspsntrim 20708 cnfldsub 20972 m2detleiblem7 22128 chpscmatgsumbin 22345 tgpconncomp 23616 tsmssub 23652 tsmsxplem1 23656 isngp4 24120 ngpsubcan 24122 ngptgp 24144 tngngp3 24172 clmpm1dir 24618 cphipval 24759 deg1suble 25624 deg1sub 25625 dchr2sum 26773 ogrpsub 32229 symgsubg 32243 cycpmconjv 32296 archiabllem2c 32336 linds2eq 32492 ressply1sub 32654 lflsub 37932 ldualvsubval 38022 lcdvsubval 40484 baerlem3lem1 40573 baerlem5alem1 40574 baerlem5amN 40582 baerlem5bmN 40583 baerlem5abmN 40584 hdmapsub 40713 nelsubgsubcld 41074 rngsubdi 46660 rngsubdir 46661 |
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