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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 2771 | . . . . . . 7 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2771 | . . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
3 | 1, 2 | lmhmf 19540 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇)) |
4 | frel 6346 | . . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → Rel 𝐹) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → Rel 𝐹) |
6 | dfrel2 5883 | . . . . 5 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
7 | 5, 6 | sylib 210 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 = 𝐹) |
8 | id 22 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
9 | 7, 8 | eqeltrd 2859 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡◡𝐹 ∈ (𝑆 LMHom 𝑇)) |
10 | 9 | anim1ci 607 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇))) |
11 | islmim2 19572 | . 2 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆))) | |
12 | islmim2 19572 | . 2 ⊢ (◡𝐹 ∈ (𝑇 LMIso 𝑆) ↔ (◡𝐹 ∈ (𝑇 LMHom 𝑆) ∧ ◡◡𝐹 ∈ (𝑆 LMHom 𝑇))) | |
13 | 10, 11, 12 | 3imtr4i 284 | 1 ⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ◡ccnv 5402 Rel wrel 5408 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 LMHom clmhm 19525 LMIso clmim 19526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-ghm 18139 df-lmod 19370 df-lmhm 19528 df-lmim 19529 |
This theorem is referenced by: lmicsym 19578 lbslcic 20702 |
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