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Theorem lmimco 21882
Description: The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.)
Assertion
Ref Expression
lmimco ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))

Proof of Theorem lmimco
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2735 . . 3 (Base‘𝑇) = (Base‘𝑇)
31, 2islmim 21079 . 2 (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4 eqid 2735 . . 3 (Base‘𝑅) = (Base‘𝑅)
54, 1islmim 21079 . 2 (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6 lmhmco 21060 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMHom 𝑇))
76ad2ant2r 747 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) ∧ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) → (𝐹𝐺) ∈ (𝑅 LMHom 𝑇))
8 f1oco 6872 . . . 4 ((𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) → (𝐹𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇))
98ad2ant2l 746 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) ∧ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) → (𝐹𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇))
104, 2islmim 21079 . . 3 ((𝐹𝐺) ∈ (𝑅 LMIso 𝑇) ↔ ((𝐹𝐺) ∈ (𝑅 LMHom 𝑇) ∧ (𝐹𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇)))
117, 9, 10sylanbrc 583 . 2 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)) ∧ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))
123, 5, 11syl2anb 598 1 ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  ccom 5693  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Basecbs 17245   LMHom clmhm 21036   LMIso clmim 21037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244  df-lmod 20877  df-lmhm 21039  df-lmim 21040
This theorem is referenced by:  lmictra  21883
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