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Definition df-lmim 18790
Description: An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
Assertion
Ref Expression
df-lmim LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
Distinct variable group:   𝑡,𝑠,𝑔

Detailed syntax breakdown of Definition df-lmim
StepHypRef Expression
1 clmim 18787 . 2 class LMIso
2 vs . . 3 setvar 𝑠
3 vt . . 3 setvar 𝑡
4 clmod 18632 . . 3 class LMod
52cv 1473 . . . . . 6 class 𝑠
6 cbs 15641 . . . . . 6 class Base
75, 6cfv 5790 . . . . 5 class (Base‘𝑠)
83cv 1473 . . . . . 6 class 𝑡
98, 6cfv 5790 . . . . 5 class (Base‘𝑡)
10 vg . . . . . 6 setvar 𝑔
1110cv 1473 . . . . 5 class 𝑔
127, 9, 11wf1o 5789 . . . 4 wff 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)
13 clmhm 18786 . . . . 5 class LMHom
145, 8, 13co 6527 . . . 4 class (𝑠 LMHom 𝑡)
1512, 10, 14crab 2899 . . 3 class {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}
162, 3, 4, 4, 15cmpt2 6529 . 2 class (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
171, 16wceq 1474 1 wff LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
Colors of variables: wff setvar class
This definition is referenced by:  lmimfn  18793  islmim  18829
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