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Mirrors > Home > MPE Home > Th. List > lmimlmhm | Structured version Visualization version GIF version |
Description: An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
lmimlmhm | ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2738 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | 1, 2 | islmim 19953 | . 2 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:(Base‘𝑅)–1-1-onto→(Base‘𝑆))) |
4 | 3 | simplbi 501 | 1 ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 –1-1-onto→wf1o 6338 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 LMHom clmhm 19910 LMIso clmim 19911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-lmhm 19913 df-lmim 19914 |
This theorem is referenced by: lmimgim 19956 lmiclcl 19961 lmicrcl 19962 lmimlbs 20652 lnmlmic 40485 |
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