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Mirrors > Home > MPE Home > Th. List > sraip | Structured version Visualization version GIF version |
Description: The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
sraip | ⊢ (𝜑 → (.r‘𝑊) = (·𝑖‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7168 | . . . 4 ⊢ ((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) ∈ V | |
2 | fvex 6658 | . . . 4 ⊢ (.r‘𝑊) ∈ V | |
3 | ipid 16634 | . . . . 5 ⊢ ·𝑖 = Slot (·𝑖‘ndx) | |
4 | 3 | setsid 16530 | . . . 4 ⊢ ((((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) ∈ V ∧ (.r‘𝑊) ∈ V) → (.r‘𝑊) = (·𝑖‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
5 | 1, 2, 4 | mp2an 691 | . . 3 ⊢ (.r‘𝑊) = (·𝑖‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
6 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
8 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
9 | sraval 19941 | . . . . . 6 ⊢ ((𝑊 ∈ V ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | |
10 | 8, 9 | sylan2 595 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
11 | 7, 10 | eqtrd 2833 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
12 | 11 | fveq2d 6649 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (·𝑖‘𝐴) = (·𝑖‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
13 | 5, 12 | eqtr4id 2852 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = (·𝑖‘𝐴)) |
14 | 3 | str0 16527 | . . 3 ⊢ ∅ = (·𝑖‘∅) |
15 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (.r‘𝑊) = ∅) | |
16 | 15 | adantr 484 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ∅) |
17 | fv2prc 6685 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
18 | 6, 17 | sylan9eqr 2855 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
19 | 18 | fveq2d 6649 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (·𝑖‘𝐴) = (·𝑖‘∅)) |
20 | 14, 16, 19 | 3eqtr4a 2859 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = (·𝑖‘𝐴)) |
21 | 13, 20 | pm2.61ian 811 | 1 ⊢ (𝜑 → (.r‘𝑊) = (·𝑖‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 sSet csts 16473 Basecbs 16475 ↾s cress 16476 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 ·𝑖cip 16562 subringAlg csra 19933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16478 df-slot 16479 df-sets 16482 df-ip 16575 df-sra 19937 |
This theorem is referenced by: frlmip 20467 rrxip 23994 |
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