Theorem lmimcnv 21354

Description: The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
lmimcnv	⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆))

Proof of Theorem lmimcnv

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	lmhmf 21321	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4		frel 6723	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
5	3, 4	syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹)
6		dfrel2 6186	. . . . 5 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
7	5, 6	sylib 221	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 = 𝐹)
8		id 23	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
9	7, 8	eqeltrd 2863	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 ∈ (𝑆 LMHom 𝑇))
10	9	anim1ci 628	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)))
11		islmim2 21353	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))
12		islmim2 21353	. 2 ⊢ (◡𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)))
13	10, 11, 12	3imtr4i 295	1 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ^◡ccnv 5653  Rel wrel 5659  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427  Basecbs 17399   LMHom clmhm 21306   LMIso clmim 21307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-map 8860  df-mgm 18828  df-sgrp 18920  df-mnd 18936  df-grp 19159  df-ghm 19440  df-lmod 21149  df-lmhm 21309  df-lmim 21310

This theorem is used by:  lmicsym  21359  lbslcic  22159