![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > lmhmlem | Structured version Visualization version GIF version |
Description: Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmhmlem.k | ⊢ 𝐾 = (Scalar‘𝑆) |
lmhmlem.l | ⊢ 𝐿 = (Scalar‘𝑇) |
Ref | Expression |
---|---|
lmhmlem | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmlem.k | . . 3 ⊢ 𝐾 = (Scalar‘𝑆) | |
2 | lmhmlem.l | . . 3 ⊢ 𝐿 = (Scalar‘𝑇) | |
3 | eqid 2824 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | eqid 2824 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
5 | eqid 2824 | . . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
6 | eqid 2824 | . . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇) | |
7 | 1, 2, 3, 4, 5, 6 | islmhm 19385 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑎 ∈ (Base‘𝐾)∀𝑏 ∈ (Base‘𝑆)(𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏))))) |
8 | 3simpa 1184 | . . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑎 ∈ (Base‘𝐾)∀𝑏 ∈ (Base‘𝑆)(𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏))) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)) | |
9 | 8 | anim2i 612 | . 2 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑎 ∈ (Base‘𝐾)∀𝑏 ∈ (Base‘𝑆)(𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾))) |
10 | 7, 9 | sylbi 209 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3116 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 Scalarcsca 16307 ·𝑠 cvsca 16308 GrpHom cghm 18007 LModclmod 19218 LMHom clmhm 19377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-iota 6085 df-fun 6124 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-lmhm 19380 |
This theorem is referenced by: lmhmsca 19388 lmghm 19389 lmhmlmod2 19390 lmhmlmod1 19391 |
Copyright terms: Public domain | W3C validator |