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Mirrors > Home > MPE Home > Th. List > lmimcnv | Structured version Visualization version GIF version |
Description: The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
lmimcnv | ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | 1, 2 | lmhmf 19799 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | frel 6492 | . . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹) |
6 | dfrel2 6013 | . . . . 5 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
7 | 5, 6 | sylib 221 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 = 𝐹) |
8 | id 22 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
9 | 7, 8 | eqeltrd 2890 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)) |
10 | 9 | anim1ci 618 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇))) |
11 | islmim2 19831 | . 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆))) | |
12 | islmim2 19831 | . 2 ⊢ (◡𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇))) | |
13 | 10, 11, 12 | 3imtr4i 295 | 1 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ◡ccnv 5518 Rel wrel 5524 ⟶wf 6320 ‘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-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-ov 7138 df-oprab 7139 df-mpo 7140 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-ghm 18348 df-lmod 19629 df-lmhm 19787 df-lmim 19788 |
This theorem is referenced by: lmicsym 19837 lbslcic 20530 |
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