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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 6900 | . . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊)) | |
2 | fveq2 6900 | . . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
3 | 1, 2 | fveq12d 6907 | . . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
4 | df-rgmod 21064 | . . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
5 | fvex 6913 | . . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V | |
6 | 3, 4, 5 | fvmpt 7008 | . 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
7 | 0fv 6944 | . . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅ | |
8 | 7 | eqcomi 2736 | . . 3 ⊢ ∅ = (∅‘(Base‘𝑊)) |
9 | fvprc 6892 | . . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅) | |
10 | fvprc 6892 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅) | |
11 | 10 | fveq1d 6902 | . . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊))) |
12 | 8, 9, 11 | 3eqtr4a 2793 | . 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 1533 ∈ wcel 2098 Vcvv 3471 ∅c0 4324 ‘cfv 6551 Basecbs 17185 subringAlg csra 21061 ringLModcrglmod 21062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-rgmod 21064 |
This theorem is referenced by: rlmval2 21090 rlmbas 21091 rlmplusg 21092 rlm0 21093 rlmmulr 21095 rlmsca 21096 rlmsca2 21097 rlmvsca 21098 rlmtopn 21099 rlmds 21100 rlmlmod 21101 frlmip 21717 rlmassa 21809 rlmnlm 24623 rlmbn 25307 rrxprds 25335 rlmdim 33312 rgmoddimOLD 33313 extdgid 33357 |
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