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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 6499 | . . . 4 ⊢ (𝑎 = 𝑊 → (subringAlg ‘𝑎) = (subringAlg ‘𝑊)) | |
2 | fveq2 6499 | . . . 4 ⊢ (𝑎 = 𝑊 → (Base‘𝑎) = (Base‘𝑊)) | |
3 | 1, 2 | fveq12d 6506 | . . 3 ⊢ (𝑎 = 𝑊 → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
4 | df-rgmod 19667 | . . 3 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
5 | fvex 6512 | . . 3 ⊢ ((subringAlg ‘𝑊)‘(Base‘𝑊)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6595 | . 2 ⊢ (𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
7 | 0fv 6539 | . . . 4 ⊢ (∅‘(Base‘𝑊)) = ∅ | |
8 | 7 | eqcomi 2787 | . . 3 ⊢ ∅ = (∅‘(Base‘𝑊)) |
9 | fvprc 6492 | . . 3 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ∅) | |
10 | fvprc 6492 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (subringAlg ‘𝑊) = ∅) | |
11 | 10 | fveq1d 6501 | . . 3 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (∅‘(Base‘𝑊))) |
12 | 8, 9, 11 | 3eqtr4a 2840 | . 2 ⊢ (¬ 𝑊 ∈ V → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
13 | 6, 12 | pm2.61i 177 | 1 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ∅c0 4178 ‘cfv 6188 Basecbs 16339 subringAlg csra 19662 ringLModcrglmod 19663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-rgmod 19667 |
This theorem is referenced by: rlmval2 19688 rlmbas 19689 rlmplusg 19690 rlm0 19691 rlmmulr 19693 rlmsca 19694 rlmsca2 19695 rlmvsca 19696 rlmtopn 19697 rlmds 19698 rlmlmod 19699 rlmassa 19820 frlmip 20624 rlmnlm 23000 rlmbn 23667 rrxprds 23695 rgmoddim 30643 extdgid 30685 |
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