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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvsca | Structured version Visualization version GIF version |
Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) |
Ref | Expression |
---|---|
dvhfvsca.h | β’ π» = (LHypβπΎ) |
dvhfvsca.t | β’ π = ((LTrnβπΎ)βπ) |
dvhfvsca.e | β’ πΈ = ((TEndoβπΎ)βπ) |
dvhfvsca.u | β’ π = ((DVecHβπΎ)βπ) |
dvhfvsca.s | β’ Β· = ( Β·π βπ) |
Ref | Expression |
---|---|
dvhvsca | β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β (π Γ πΈ))) β (π Β· πΉ) = β¨(π β(1st βπΉ)), (π β (2nd βπΉ))β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhfvsca.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | dvhfvsca.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | dvhfvsca.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
4 | dvhfvsca.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
5 | dvhfvsca.s | . . . 4 β’ Β· = ( Β·π βπ) | |
6 | 1, 2, 3, 4, 5 | dvhfvsca 40577 | . . 3 β’ ((πΎ β π β§ π β π») β Β· = (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)) |
7 | 6 | oveqd 7441 | . 2 β’ ((πΎ β π β§ π β π») β (π Β· πΉ) = (π (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)πΉ)) |
8 | eqid 2727 | . . 3 β’ (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©) = (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©) | |
9 | 8 | dvhvscaval 40576 | . 2 β’ ((π β πΈ β§ πΉ β (π Γ πΈ)) β (π (π β πΈ, π β (π Γ πΈ) β¦ β¨(π β(1st βπ)), (π β (2nd βπ))β©)πΉ) = β¨(π β(1st βπΉ)), (π β (2nd βπΉ))β©) |
10 | 7, 9 | sylan9eq 2787 | 1 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β (π Γ πΈ))) β (π Β· πΉ) = β¨(π β(1st βπΉ)), (π β (2nd βπΉ))β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β¨cop 4636 Γ cxp 5678 β ccom 5684 βcfv 6551 (class class class)co 7424 β cmpo 7426 1st c1st 7995 2nd c2nd 7996 Β·π cvsca 17242 LHypclh 39461 LTrncltrn 39578 TEndoctendo 40229 DVecHcdvh 40555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-struct 17121 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-sca 17254 df-vsca 17255 df-dvech 40556 |
This theorem is referenced by: dvhopvsca 40579 dvhvscacl 40580 dvhlveclem 40585 diblss 40647 dicvscacl 40668 |
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