Theorem lmimco 21803

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 2737	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2737	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	islmim 21018	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4		eqid 2737	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4, 1	islmim 21018	. 2 ⊢ (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6		lmhmco 20999	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMHom 𝑇))
7	6	ad2ant2r 748	. . 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 6798	. . . 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 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))) → (𝐹 ∘ 𝐺):(Base‘𝑅)–1-1-onto→(Base‘𝑇))
10	4, 2	islmim 21018	. . 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 584	. 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 599	1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ∘ ccom 5629  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   LMHom clmhm 20975   LMIso clmim 20976

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-0g 17365  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-mhm 18712  df-grp 18870  df-ghm 19146  df-lmod 20817  df-lmhm 20978  df-lmim 20979

This theorem is referenced by:  lmictra  21804