Theorem lmimco 19940

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)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))

Proof of Theorem lmimco

Step	Hyp	Ref	Expression
1		eqid 2605	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2605	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	islmim 18825	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4		eqid 2605	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4, 1	islmim 18825	. 2 ⊢ (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6		lmhmco 18806	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMHom 𝑇))
7	6	ad2ant2r 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))) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMHom 𝑇))
8		f1oco 6053	. . . 4 ⊢ ((𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) → (𝐹 ∘ 𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇))
9	8	ad2ant2l 777	. . 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))) → (𝐹 ∘ 𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇))
10	4, 2	islmim 18825	. . 3 ⊢ ((𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇) ↔ ((𝐹 ∘ 𝐺) ∈ (𝑅 LMHom 𝑇) ∧ (𝐹 ∘ 𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇)))
11	7, 9, 10	sylanbrc 694	. 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))) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))
12	3, 5, 11	syl2anb 494	1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))

Syntax hints:   → wi 4   ∧ wa 382   ∈ wcel 1975   ∘ ccom 5028  –1-1-onto→wf1o 5785  ‘cfv 5786  (class class class)co 6523  Basecbs 15637   LMHom clmhm 18782   LMIso clmim 18783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-map 7719  df-0g 15867  df-mgm 17007  df-sgrp 17049  df-mnd 17060  df-mhm 17100  df-grp 17190  df-ghm 17423  df-lmod 18630  df-lmhm 18785  df-lmim 18786

This theorem is referenced by:  lmictra  19941