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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 6191 | . . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊)) | |
2 | fveq2 6191 | . . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
3 | 1, 2 | fveq12d 6197 | . . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
4 | df-rgmod 19173 | . . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
5 | fvex 6201 | . . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6282 | . 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
7 | 0fv 6227 | . . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅ | |
8 | 7 | eqcomi 2631 | . . 3 ⊢ ∅ = (∅‘(Base‘𝑊)) |
9 | fvprc 6185 | . . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅) | |
10 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅) | |
11 | 10 | fveq1d 6193 | . . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊))) |
12 | 8, 9, 11 | 3eqtr4a 2682 | . 2 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
13 | 6, 12 | pm2.61i 176 | 1 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ‘cfv 5888 Basecbs 15857 subringAlg csra 19168 ringLModcrglmod 19169 |
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-8 1992 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-pow 4843 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-rgmod 19173 |
This theorem is referenced by: rlmval2 19194 rlmbas 19195 rlmplusg 19196 rlm0 19197 rlmmulr 19199 rlmsca 19200 rlmsca2 19201 rlmvsca 19202 rlmtopn 19203 rlmds 19204 rlmlmod 19205 rlmassa 19326 frlmip 20117 rlmnlm 22492 rlmbn 23157 rrxprds 23177 |
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