![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sravsca | Structured version Visualization version GIF version |
Description: The scalar product operation 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 |
---|---|
sravsca | ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | . . . . . 6 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
2 | 1 | adantl 474 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
3 | srapart.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
4 | sraval 19660 | . . . . . 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)〉)) | |
5 | 3, 4 | sylan2 583 | . . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
6 | 2, 5 | eqtrd 2808 | . . . 4 ⊢ ((𝑊 ∈ V ∧ 𝜑) → 𝐴 = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
7 | 6 | fveq2d 6497 | . . 3 ⊢ ((𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉))) |
8 | ovex 7002 | . . . . 5 ⊢ (𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V | |
9 | fvex 6506 | . . . . 5 ⊢ (.r‘𝑊) ∈ V | |
10 | vscaid 16481 | . . . . . 6 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx) | |
11 | 10 | setsid 16384 | . . . . 5 ⊢ (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) ∈ V ∧ (.r‘𝑊) ∈ V) → (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉))) |
12 | 8, 9, 11 | mp2an 679 | . . . 4 ⊢ (.r‘𝑊) = ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) |
13 | 6re 11526 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
14 | 6lt8 11633 | . . . . . . 7 ⊢ 6 < 8 | |
15 | 13, 14 | ltneii 10545 | . . . . . 6 ⊢ 6 ≠ 8 |
16 | vscandx 16480 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
17 | ipndx 16487 | . . . . . . 7 ⊢ (·𝑖‘ndx) = 8 | |
18 | 16, 17 | neeq12i 3027 | . . . . . 6 ⊢ (( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) ↔ 6 ≠ 8) |
19 | 15, 18 | mpbir 223 | . . . . 5 ⊢ ( ·𝑠 ‘ndx) ≠ (·𝑖‘ndx) |
20 | 10, 19 | setsnid 16385 | . . . 4 ⊢ ( ·𝑠 ‘((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉)) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
21 | 12, 20 | eqtri 2796 | . . 3 ⊢ (.r‘𝑊) = ( ·𝑠 ‘(((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) |
22 | 7, 21 | syl6reqr 2827 | . 2 ⊢ ((𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
23 | 10 | str0 16381 | . . 3 ⊢ ∅ = ( ·𝑠 ‘∅) |
24 | fvprc 6486 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (.r‘𝑊) = ∅) | |
25 | 24 | adantr 473 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ∅) |
26 | fv2prc 6534 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → ((subringAlg ‘𝑊)‘𝑆) = ∅) | |
27 | 1, 26 | sylan9eqr 2830 | . . . 4 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → 𝐴 = ∅) |
28 | 27 | fveq2d 6497 | . . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘∅)) |
29 | 23, 25, 28 | 3eqtr4a 2834 | . 2 ⊢ ((¬ 𝑊 ∈ V ∧ 𝜑) → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
30 | 22, 29 | pm2.61ian 799 | 1 ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 Vcvv 3409 ⊆ wss 3825 ∅c0 4173 〈cop 4441 ‘cfv 6182 (class class class)co 6970 6c6 11492 8c8 11494 ndxcnx 16326 sSet csts 16327 Basecbs 16329 ↾s cress 16330 .rcmulr 16412 Scalarcsca 16414 ·𝑠 cvsca 16415 ·𝑖cip 16416 subringAlg csra 19652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-ndx 16332 df-slot 16333 df-sets 16336 df-vsca 16428 df-ip 16429 df-sra 19656 |
This theorem is referenced by: sralmod 19671 rlmvsca 19686 sraassa 19809 sranlm 22986 drgextvsca 30578 drgextlsp 30581 fedgmullem1 30610 extdg1id 30638 ccfldsrarelvec 30641 ccfldextdgrr 30642 |
Copyright terms: Public domain | W3C validator |