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Mirrors > Home > MPE Home > Th. List > rlmval2 | Structured version Visualization version GIF version |
Description: Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
rlmval2 | ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet 〈(Scalar‘ndx), 𝑊〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmval 20461 | . . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
3 | ssid 3943 | . . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊) | |
4 | sraval 20438 | . . 3 ⊢ ((𝑊 ∈ 𝑋 ∧ (Base‘𝑊) ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | |
5 | 3, 4 | mpan2 688 | . 2 ⊢ (𝑊 ∈ 𝑋 → ((subringAlg ‘𝑊)‘(Base‘𝑊)) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
6 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | 6 | ressid 16954 | . . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊) |
8 | 7 | opeq2d 4811 | . . . . 5 ⊢ (𝑊 ∈ 𝑋 → 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉 = 〈(Scalar‘ndx), 𝑊〉) |
9 | 8 | oveq2d 7291 | . . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) = (𝑊 sSet 〈(Scalar‘ndx), 𝑊〉)) |
10 | 9 | oveq1d 7290 | . . 3 ⊢ (𝑊 ∈ 𝑋 → ((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) = ((𝑊 sSet 〈(Scalar‘ndx), 𝑊〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
11 | 10 | oveq1d 7290 | . 2 ⊢ (𝑊 ∈ 𝑋 → (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉) = (((𝑊 sSet 〈(Scalar‘ndx), 𝑊〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
12 | 2, 5, 11 | 3eqtrd 2782 | 1 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet 〈(Scalar‘ndx), 𝑊〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 〈cop 4567 ‘cfv 6433 (class class class)co 7275 sSet csts 16864 ndxcnx 16894 Basecbs 16912 ↾s cress 16941 .rcmulr 16963 Scalarcsca 16965 ·𝑠 cvsca 16966 ·𝑖cip 16967 subringAlg csra 20430 ringLModcrglmod 20431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-ress 16942 df-sra 20434 df-rgmod 20435 |
This theorem is referenced by: (None) |
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