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Mirrors > Home > MPE Home > Th. List > islmhm3 | Structured version Visualization version GIF version |
Description: Property of a module homomorphism, similar to ismhm 17952. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
islmhm.k | ⊢ 𝐾 = (Scalar‘𝑆) |
islmhm.l | ⊢ 𝐿 = (Scalar‘𝑇) |
islmhm.b | ⊢ 𝐵 = (Base‘𝐾) |
islmhm.e | ⊢ 𝐸 = (Base‘𝑆) |
islmhm.m | ⊢ · = ( ·𝑠 ‘𝑆) |
islmhm.n | ⊢ × = ( ·𝑠 ‘𝑇) |
Ref | Expression |
---|---|
islmhm3 | ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islmhm.k | . . 3 ⊢ 𝐾 = (Scalar‘𝑆) | |
2 | islmhm.l | . . 3 ⊢ 𝐿 = (Scalar‘𝑇) | |
3 | islmhm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
4 | islmhm.e | . . 3 ⊢ 𝐸 = (Base‘𝑆) | |
5 | islmhm.m | . . 3 ⊢ · = ( ·𝑠 ‘𝑆) | |
6 | islmhm.n | . . 3 ⊢ × = ( ·𝑠 ‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | islmhm 19793 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
8 | 7 | baib 538 | 1 ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 GrpHom cghm 18349 LModclmod 19628 LMHom clmhm 19785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-iota 6309 df-fun 6352 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-lmhm 19788 |
This theorem is referenced by: islmhm2 19804 pj1lmhm 19866 |
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