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Theorem lmimco 20002
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 2610 . . 3 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2610 . . 3 (Base‘𝑇) = (Base‘𝑇)
31, 2islmim 18883 . 2 (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4 eqid 2610 . . 3 (Base‘𝑅) = (Base‘𝑅)
54, 1islmim 18883 . 2 (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6 lmhmco 18864 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMHom 𝑇))
76ad2ant2r 779 . . 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 6072 . . . 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 778 . . 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 18883 . . 3 ((𝐹𝐺) ∈ (𝑅 LMIso 𝑇) ↔ ((𝐹𝐺) ∈ (𝑅 LMHom 𝑇) ∧ (𝐹𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇)))
117, 9, 10sylanbrc 695 . 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 495 1 ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 1977  ccom 5042  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  Basecbs 15695   LMHom clmhm 18840   LMIso clmim 18841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-lmod 18688  df-lmhm 18843  df-lmim 18844
This theorem is referenced by:  lmictra  20003
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