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 20075 | . . 3 ⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))) |
3 | ssid 3897 | . . 3 ⊢ (Base‘𝑊) ⊆ (Base‘𝑊) | |
4 | sraval 20060 | . . 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 691 | . 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 16655 | . . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (𝑊 ↾s (Base‘𝑊)) = 𝑊) |
8 | 7 | opeq2d 4765 | . . . . 5 ⊢ (𝑊 ∈ 𝑋 → 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉 = 〈(Scalar‘ndx), 𝑊〉) |
9 | 8 | oveq2d 7180 | . . . 4 ⊢ (𝑊 ∈ 𝑋 → (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s (Base‘𝑊))〉) = (𝑊 sSet 〈(Scalar‘ndx), 𝑊〉)) |
10 | 9 | oveq1d 7179 | . . 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 7179 | . 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 2777 | 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 1542 ∈ wcel 2113 ⊆ wss 3841 〈cop 4519 ‘cfv 6333 (class class class)co 7164 ndxcnx 16576 sSet csts 16577 Basecbs 16579 ↾s cress 16580 .rcmulr 16662 Scalarcsca 16664 ·𝑠 cvsca 16665 ·𝑖cip 16666 subringAlg csra 20052 ringLModcrglmod 20053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-ress 16587 df-sra 20056 df-rgmod 20057 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |