![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rlmval | Structured version Visualization version GIF version |
Description: Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
rlmval | ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊)) | |
2 | fveq2 6907 | . . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
3 | 1, 2 | fveq12d 6914 | . . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
4 | df-rgmod 21191 | . . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
5 | fvex 6920 | . . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V | |
6 | 3, 4, 5 | fvmpt 7016 | . 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
7 | 0fv 6951 | . . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅ | |
8 | 7 | eqcomi 2744 | . . 3 ⊢ ∅ = (∅‘(Base‘𝑊)) |
9 | fvprc 6899 | . . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅) | |
10 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅) | |
11 | 10 | fveq1d 6909 | . . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊))) |
12 | 8, 9, 11 | 3eqtr4a 2801 | . 2 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
13 | 6, 12 | pm2.61i 182 | 1 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ‘cfv 6563 Basecbs 17245 subringAlg csra 21188 ringLModcrglmod 21189 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-rgmod 21191 |
This theorem is referenced by: rlmval2 21217 rlmbas 21218 rlmplusg 21219 rlm0 21220 rlmmulr 21222 rlmsca 21223 rlmsca2 21224 rlmvsca 21225 rlmtopn 21226 rlmds 21227 rlmlmod 21228 frlmip 21816 rlmassa 21909 rlmnlm 24725 rlmbn 25409 rrxprds 25437 rlmdim 33637 rgmoddimOLD 33638 extdgid 33688 |
Copyright terms: Public domain | W3C validator |