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Mirrors > Home > MPE Home > Th. List > islmim2 | Structured version Visualization version GIF version |
Description: An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
islmim2 | ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 LMHom 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | 1, 2 | islmim 20818 | . 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) |
4 | 1, 2 | lmhmf1o 20802 | . . 3 ⊢ (𝐹 ∈ (𝑅 LMHom 𝑆) → (𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆) ↔ ◡𝐹 ∈ (𝑆 LMHom 𝑅))) |
5 | 4 | pm5.32i 574 | . 2 ⊢ ((𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆)) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 LMHom 𝑅))) |
6 | 3, 5 | bitri 275 | 1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 LMHom 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2105 ◡ccnv 5675 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 Basecbs 17149 LMHom clmhm 20775 LMIso clmim 20776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-ghm 19129 df-lmod 20617 df-lmhm 20778 df-lmim 20779 |
This theorem is referenced by: lmimcnv 20823 lnmlmic 42133 |
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