Theorem lmimco 21975

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 2763	. . 3 ⊢ (Base‘𝑆) = (Base‘𝑆)
2		eqid 2763	. . 3 ⊢ (Base‘𝑇) = (Base‘𝑇)
3	1, 2	islmim 21164	. 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:(Base‘𝑆)–1-1-onto→(Base‘𝑇)))
4		eqid 2763	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	4, 1	islmim 21164	. 2 ⊢ (𝐺 ∈ (𝑅 LMIso 𝑆) ↔ (𝐺 ∈ (𝑅 LMHom 𝑆) ∧ 𝐺:(Base‘𝑅)–1-1-onto→(Base‘𝑆)))
6		lmhmco 21145	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑅 LMHom 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMHom 𝑇))
7	6	ad2ant2r 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) ∈ (𝑅 LMHom 𝑇))
8		f1oco 6846	. . . 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 758	. . 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 21164	. . 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 594	. 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 609	1 ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143   ∘ ccom 5667  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Basecbs 17270   LMHom clmhm 21121   LMIso clmim 21122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-grp 19004  df-ghm 19285  df-lmod 20964  df-lmhm 21124  df-lmim 21125

This theorem is referenced by:  lmictra  21976