Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19191 | . . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
3 | ssid 3624 | . . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊) | |
4 | sraval 19176 | . . 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 707 | . 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 2622 | . . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
7 | 6 | ressid 15935 | . . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊) |
8 | 7 | opeq2d 4409 | . . . . 5 ⊢ (𝑊 ∈ 𝑋 → 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉 = 〈(Scalar‘ndx), 𝑊〉) |
9 | 8 | oveq2d 6666 | . . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) = (𝑊 sSet 〈(Scalar‘ndx), 𝑊〉)) |
10 | 9 | oveq1d 6665 | . . 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 6665 | . 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 2660 | 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 1483 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 ‘cfv 5888 (class class class)co 6650 ndxcnx 15854 sSet csts 15855 Basecbs 15857 ↾s cress 15858 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 ·𝑖cip 15946 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-rep 4771 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-ress 15865 df-sra 19172 df-rgmod 19173 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |