Theorem lmimcnv 21188

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 21155	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4		frel 6711	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
5	3, 4	syl 18	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹)
6		dfrel2 6187	. . . . 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 627	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)))
11		islmim2 21187	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))
12		islmim2 21187	. 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))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ^◡ccnv 5660  Rel wrel 5666  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17264   LMHom clmhm 21140   LMIso clmim 21141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-ghm 19279  df-lmod 20983  df-lmhm 21143  df-lmim 21144

This theorem is referenced by:  lmicsym  21193  lbslcic  21991