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Mirrors > Home > MPE Home > Th. List > reldmlmhm | Structured version Visualization version GIF version |
Description: Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
reldmlmhm | ⊢ Rel dom LMHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-lmhm 19022 | . 2 ⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))}) | |
2 | 1 | reldmmpt2 6771 | 1 ⊢ Rel dom LMHom |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∀wral 2912 {crab 2916 [wsbc 3435 dom cdm 5114 Rel wrel 5119 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 GrpHom cghm 17657 LModclmod 18863 LMHom clmhm 19019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-dm 5124 df-oprab 6654 df-mpt2 6655 df-lmhm 19022 |
This theorem is referenced by: mendbas 37754 |
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