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Mirrors > Home > MPE Home > Th. List > grpolinv | Structured version Visualization version GIF version |
Description: The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinv.1 | ⊢ 𝑋 = ran 𝐺 |
grpinv.2 | ⊢ 𝑈 = (GId‘𝐺) |
grpinv.3 | ⊢ 𝑁 = (inv‘𝐺) |
Ref | Expression |
---|---|
grpolinv | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinv.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
2 | grpinv.2 | . . 3 ⊢ 𝑈 = (GId‘𝐺) | |
3 | grpinv.3 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
4 | 1, 2, 3 | grpoinv 27956 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑁‘𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁‘𝐴)) = 𝑈)) |
5 | 4 | simpld 490 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴)𝐺𝐴) = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ran crn 5358 ‘cfv 6137 (class class class)co 6924 GrpOpcgr 27920 GIdcgi 27921 invcgn 27922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-grpo 27924 df-gid 27925 df-ginv 27926 |
This theorem is referenced by: grpoinvid1 27959 grpoinvid2 27960 grpolcan 27961 grpo2inv 27962 grponpcan 27974 nvlinv 28083 hhssabloilem 28694 rngoaddneg2 34357 isdrngo2 34386 |
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