Theorem lmimcnv 21019

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 2736	. . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	lmhmf 20986	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
4		frel 6667	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹)
6		dfrel2 6147	. . . . 5 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
7	5, 6	sylib 218	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 = 𝐹)
8		id 22	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
9	7, 8	eqeltrd 2836	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 ∈ (𝑆 LMHom 𝑇))
10	9	anim1ci 616	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)))
11		islmim2 21018	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))
12		islmim2 21018	. 2 ⊢ (◡𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)))
13	10, 11, 12	3imtr4i 292	1 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ^◡ccnv 5623  Rel wrel 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136   LMHom clmhm 20971   LMIso clmim 20972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-ghm 19142  df-lmod 20813  df-lmhm 20974  df-lmim 20975

This theorem is referenced by:  lmicsym  21024  lbslcic  21796