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Theorem lmimco 20533
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 2798 . . 3 (Base‘𝑆) = (Base‘𝑆)
2 eqid 2798 . . 3 (Base‘𝑇) = (Base‘𝑇)
31, 2islmim 19827 . 2 (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4 eqid 2798 . . 3 (Base‘𝑅) = (Base‘𝑅)
54, 1islmim 19827 . 2 (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6 lmhmco 19808 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMHom 𝑇))
76ad2ant2r 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) ∈ (𝑅 LMHom 𝑇))
8 f1oco 6612 . . . 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 745 . . 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 19827 . . 3 ((𝐹𝐺) ∈ (𝑅 LMIso 𝑇) ↔ ((𝐹𝐺) ∈ (𝑅 LMHom 𝑇) ∧ (𝐹𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇)))
117, 9, 10sylanbrc 586 . 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 600 1 ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2111  ccom 5523  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  Basecbs 16475   LMHom clmhm 19784   LMIso clmim 19785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-lmod 19629  df-lmhm 19787  df-lmim 19788
This theorem is referenced by:  lmictra  20534
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