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| 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 6905 | . . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊)) | |
| 2 | fveq2 6905 | . . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
| 3 | 1, 2 | fveq12d 6912 | . . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) | 
| 4 | df-rgmod 21174 | . . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
| 5 | fvex 6918 | . . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V | |
| 6 | 3, 4, 5 | fvmpt 7015 | . 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) | 
| 7 | 0fv 6949 | . . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅ | |
| 8 | 7 | eqcomi 2745 | . . 3 ⊢ ∅ = (∅‘(Base‘𝑊)) | 
| 9 | fvprc 6897 | . . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅) | |
| 10 | fvprc 6897 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅) | |
| 11 | 10 | fveq1d 6907 | . . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊))) | 
| 12 | 8, 9, 11 | 3eqtr4a 2802 | . 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 1539 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 ‘cfv 6560 Basecbs 17248 subringAlg csra 21171 ringLModcrglmod 21172 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-rgmod 21174 | 
| This theorem is referenced by: rlmval2 21200 rlmbas 21201 rlmplusg 21202 rlm0 21203 rlmmulr 21205 rlmsca 21206 rlmsca2 21207 rlmvsca 21208 rlmtopn 21209 rlmds 21210 rlmlmod 21211 frlmip 21799 rlmassa 21892 rlmnlm 24710 rlmbn 25396 rrxprds 25424 rlmdim 33661 rgmoddimOLD 33662 extdgid 33712 | 
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