Theorem lmimco 22130

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 2761	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2761	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	islmim 21317	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4		eqid 2761	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4, 1	islmim 21317	. 2 ⊢ (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6		lmhmco 21298	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMHom 𝑇))
7	6	ad2ant2r 760	. . 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 6840	. . . 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 759	. . 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 21317	. . 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 595	. 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 610	1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ∘ ccom 5655  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412  Basecbs 17367   LMHom clmhm 21274   LMIso clmim 21275

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-map 8833  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-grp 19127  df-ghm 19408  df-lmod 21117  df-lmhm 21277  df-lmim 21278

This theorem is used by:  lmictra  22131