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Mirrors > Home > MPE Home > Th. List > srasca | Structured version Visualization version GIF version |
Description: The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
srasca | ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scaid 16625 | . . . . 5 ⊢ Scalar = Slot (Scalar‘ndx) | |
2 | 5re 11712 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
3 | 5lt6 11806 | . . . . . . 7 ⊢ 5 < 6 | |
4 | 2, 3 | ltneii 10742 | . . . . . 6 ⊢ 5 ≠ 6 |
5 | scandx 16624 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
6 | vscandx 16626 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
7 | 5, 6 | neeq12i 3053 | . . . . . 6 ⊢ ((Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ 5 ≠ 6) |
8 | 4, 7 | mpbir 234 | . . . . 5 ⊢ (Scalar‘ndx) ≠ ( ·𝑠 ‘ndx) |
9 | 1, 8 | setsnid 16531 | . . . 4 ⊢ (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) = (Scalar‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
10 | 5lt8 11819 | . . . . . . 7 ⊢ 5 < 8 | |
11 | 2, 10 | ltneii 10742 | . . . . . 6 ⊢ 5 ≠ 8 |
12 | ipndx 16633 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
13 | 5, 12 | neeq12i 3053 | . . . . . 6 ⊢ ((Scalar‘ndx) ≠ (·𝑖‘ndx) ↔ 5 ≠ 8) |
14 | 11, 13 | mpbir 234 | . . . . 5 ⊢ (Scalar‘ndx) ≠ (·𝑖‘ndx) |
15 | 1, 14 | setsnid 16531 | . . . 4 ⊢ (Scalar‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
16 | 9, 15 | eqtri 2821 | . . 3 ⊢ (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉)) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
17 | ovexd 7170 | . . . 4 ⊢ (𝜑 → (𝑊 ↾s 𝑆) ∈ V) | |
18 | 1 | setsid 16530 | . . . 4 ⊢ ((𝑊 ∈ V ∧ (𝑊 ↾s 𝑆) ∈ V) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉))) |
19 | 17, 18 | sylan2 595 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘(𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉))) |
20 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
21 | 20 | adantl 485 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
22 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
23 | 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)〉)) | |
24 | 22, 23 | 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)〉)) |
25 | 21, 24 | eqtrd 2833 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
26 | 25 | fveq2d 6649 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (Scalar‘𝐴) = (Scalar‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
27 | 16, 19, 26 | 3eqtr4a 2859 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
28 | 1 | str0 16527 | . . 3 ⊢ ∅ = (Scalar‘∅) |
29 | reldmress 16542 | . . . . 5 ⊢ Rel dom ↾s | |
30 | 29 | ovprc1 7174 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝑆) = ∅) |
31 | 30 | adantr 484 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = ∅) |
32 | fv2prc 6685 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
33 | 20, 32 | sylan9eqr 2855 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
34 | 33 | fveq2d 6649 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (Scalar‘𝐴) = (Scalar‘∅)) |
35 | 28, 31, 34 | 3eqtr4a 2859 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
36 | 27, 35 | pm2.61ian 811 | 1 ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 5c5 11683 6c6 11684 8c8 11686 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-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
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-nel 3092 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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 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-ress 16483 df-sca 16573 df-vsca 16574 df-ip 16575 df-sra 19937 |
This theorem is referenced by: sralmod 19952 rlmsca 19965 rlmsca2 19966 frlmip 20467 sraassa 20556 sranlm 23290 srabn 23964 rrxprds 23993 sralvec 31078 drgext0gsca 31082 drgextlsp 31084 fedgmullem1 31113 fedgmullem2 31114 fedgmul 31115 extdg1id 31141 ccfldsrarelvec 31144 ccfldextdgrr 31145 |
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