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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.) (Proof shortened by AV, 12-Nov-2024.) |
Ref | Expression |
---|---|
srapart.a | β’ (π β π΄ = ((subringAlg βπ)βπ)) |
srapart.s | β’ (π β π β (Baseβπ)) |
Ref | Expression |
---|---|
sravsca | β’ (π β (.rβπ) = ( Β·π βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7394 | . . . . 5 β’ (π sSet β¨(Scalarβndx), (π βΎs π)β©) β V | |
2 | fvex 6859 | . . . . 5 β’ (.rβπ) β V | |
3 | vscaid 17209 | . . . . . 6 β’ Β·π = Slot ( Β·π βndx) | |
4 | 3 | setsid 17088 | . . . . 5 β’ (((π sSet β¨(Scalarβndx), (π βΎs π)β©) β V β§ (.rβπ) β V) β (.rβπ) = ( Β·π β((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©))) |
5 | 1, 2, 4 | mp2an 691 | . . . 4 β’ (.rβπ) = ( Β·π β((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©)) |
6 | slotsdifipndx 17224 | . . . . . 6 β’ (( Β·π βndx) β (Β·πβndx) β§ (Scalarβndx) β (Β·πβndx)) | |
7 | 6 | simpli 485 | . . . . 5 β’ ( Β·π βndx) β (Β·πβndx) |
8 | 3, 7 | setsnid 17089 | . . . 4 β’ ( Β·π β((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©)) = ( Β·π β(((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
9 | 5, 8 | eqtri 2761 | . . 3 β’ (.rβπ) = ( Β·π β(((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
10 | srapart.a | . . . . . 6 β’ (π β π΄ = ((subringAlg βπ)βπ)) | |
11 | 10 | adantl 483 | . . . . 5 β’ ((π β V β§ π) β π΄ = ((subringAlg βπ)βπ)) |
12 | srapart.s | . . . . . 6 β’ (π β π β (Baseβπ)) | |
13 | sraval 20682 | . . . . . 6 β’ ((π β V β§ π β (Baseβπ)) β ((subringAlg βπ)βπ) = (((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) | |
14 | 12, 13 | sylan2 594 | . . . . 5 β’ ((π β V β§ π) β ((subringAlg βπ)βπ) = (((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
15 | 11, 14 | eqtrd 2773 | . . . 4 β’ ((π β V β§ π) β π΄ = (((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©)) |
16 | 15 | fveq2d 6850 | . . 3 β’ ((π β V β§ π) β ( Β·π βπ΄) = ( Β·π β(((π sSet β¨(Scalarβndx), (π βΎs π)β©) sSet β¨( Β·π βndx), (.rβπ)β©) sSet β¨(Β·πβndx), (.rβπ)β©))) |
17 | 9, 16 | eqtr4id 2792 | . 2 β’ ((π β V β§ π) β (.rβπ) = ( Β·π βπ΄)) |
18 | 3 | str0 17069 | . . 3 β’ β = ( Β·π ββ ) |
19 | fvprc 6838 | . . . 4 β’ (Β¬ π β V β (.rβπ) = β ) | |
20 | 19 | adantr 482 | . . 3 β’ ((Β¬ π β V β§ π) β (.rβπ) = β ) |
21 | fv2prc 6891 | . . . . 5 β’ (Β¬ π β V β ((subringAlg βπ)βπ) = β ) | |
22 | 10, 21 | sylan9eqr 2795 | . . . 4 β’ ((Β¬ π β V β§ π) β π΄ = β ) |
23 | 22 | fveq2d 6850 | . . 3 β’ ((Β¬ π β V β§ π) β ( Β·π βπ΄) = ( Β·π ββ )) |
24 | 18, 20, 23 | 3eqtr4a 2799 | . 2 β’ ((Β¬ π β V β§ π) β (.rβπ) = ( Β·π βπ΄)) |
25 | 17, 24 | pm2.61ian 811 | 1 β’ (π β (.rβπ) = ( Β·π βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 Vcvv 3447 β wss 3914 β c0 4286 β¨cop 4596 βcfv 6500 (class class class)co 7361 sSet csts 17043 ndxcnx 17073 Basecbs 17091 βΎs cress 17120 .rcmulr 17142 Scalarcsca 17144 Β·π cvsca 17145 Β·πcip 17146 subringAlg csra 20674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-sets 17044 df-slot 17062 df-ndx 17074 df-sca 17157 df-vsca 17158 df-ip 17159 df-sra 20678 |
This theorem is referenced by: sralmod 20701 rlmvsca 20716 sraassa 21296 sranlm 24071 drgextvsca 32354 drgextlsp 32357 fedgmullem1 32388 extdg1id 32416 ccfldsrarelvec 32419 ccfldextdgrr 32420 |
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