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Theorem islmim2 20956
Description: An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
islmim2 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑅)))

Proof of Theorem islmim2
StepHypRef Expression
1 eqid 2727 . . 3 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2727 . . 3 (Base‘𝑆) = (Base‘𝑆)
31, 2islmim 20952 . 2 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
41, 2lmhmf1o 20936 . . 3 (𝐹 ∈ (𝑅 LMHom 𝑆) → (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆) ↔ 𝐹 ∈ (𝑆 LMHom 𝑅)))
54pm5.32i 573 . 2 ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑅)))
63, 5bitri 274 1 (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wcel 2098  ccnv 5679  1-1-ontowf1o 6550  cfv 6551  (class class class)co 7424  Basecbs 17185   LMHom clmhm 20909   LMIso clmim 20910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18898  df-ghm 19173  df-lmod 20750  df-lmhm 20912  df-lmim 20913
This theorem is referenced by:  lmimcnv  20957  lnmlmic  42515
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