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Metamath Proof Explorer

Theorem diaf1oN

Description: The partial isomorphism A for a lattice K is a one-to-one, onto function. Part of Lemma M of Crawley p. 121 line 12, with closed subspaces rather than subspaces. See diadm for the domain. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dvadia.h ⊢ H = LHyp ⁡ K
dvadia.u ⊢ U = DVecA ⁡ K ⁡ W
dvadia.i ⊢ I = DIsoA ⁡ K ⁡ W
dvadia.n ⊢ ⊥ ˙ = ocA ⁡ K ⁡ W
dvadia.s ⊢ S = LSubSp ⁡ U
Assertion diaf1oN ⊢ K ∈ HL ∧ W ∈ H → I : dom ⁡ I ⟶ 1-1 onto x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x

Proof

Step Hyp Ref Expression
1 dvadia.h ⊢ H = LHyp ⁡ K
2 dvadia.u ⊢ U = DVecA ⁡ K ⁡ W
3 dvadia.i ⊢ I = DIsoA ⁡ K ⁡ W
4 dvadia.n ⊢ ⊥ ˙ = ocA ⁡ K ⁡ W
5 dvadia.s ⊢ S = LSubSp ⁡ U
6 1 3 diaf11N ⊢ K ∈ HL ∧ W ∈ H → I : dom ⁡ I ⟶ 1-1 onto ran ⁡ I
7 f1of1 ⊢ I : dom ⁡ I ⟶ 1-1 onto ran ⁡ I → I : dom ⁡ I ⟶ 1-1 ran ⁡ I
8 6 7 syl ⊢ K ∈ HL ∧ W ∈ H → I : dom ⁡ I ⟶ 1-1 ran ⁡ I
9 1 2 3 4 5 diarnN ⊢ K ∈ HL ∧ W ∈ H → ran ⁡ I = x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x
10 f1eq3 ⊢ ran ⁡ I = x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x → I : dom ⁡ I ⟶ 1-1 ran ⁡ I ↔ I : dom ⁡ I ⟶ 1-1 x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x
11 9 10 syl ⊢ K ∈ HL ∧ W ∈ H → I : dom ⁡ I ⟶ 1-1 ran ⁡ I ↔ I : dom ⁡ I ⟶ 1-1 x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x
12 8 11 mpbid ⊢ K ∈ HL ∧ W ∈ H → I : dom ⁡ I ⟶ 1-1 x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x
13 dff1o5 ⊢ I : dom ⁡ I ⟶ 1-1 onto x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x ↔ I : dom ⁡ I ⟶ 1-1 x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x ∧ ran ⁡ I = x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x
14 12 9 13 sylanbrc ⊢ K ∈ HL ∧ W ∈ H → I : dom ⁡ I ⟶ 1-1 onto x ∈ S | ⊥ ˙ ⁡ ⊥ ˙ ⁡ x = x