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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhopvsca | Structured version Visualization version GIF version |
Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.) |
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 |
---|---|
dvhopvsca | β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (π Β· β¨πΉ, πβ©) = β¨(π βπΉ), (π β π)β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (πΎ β π β§ π β π»)) | |
2 | simpr1 1195 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β π β πΈ) | |
3 | simpr2 1196 | . . . 4 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β πΉ β π) | |
4 | simpr3 1197 | . . . 4 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β π β πΈ) | |
5 | opelxpi 5671 | . . . 4 β’ ((πΉ β π β§ π β πΈ) β β¨πΉ, πβ© β (π Γ πΈ)) | |
6 | 3, 4, 5 | syl2anc 585 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β β¨πΉ, πβ© β (π Γ πΈ)) |
7 | dvhfvsca.h | . . . 4 β’ π» = (LHypβπΎ) | |
8 | dvhfvsca.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
9 | dvhfvsca.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
10 | dvhfvsca.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
11 | dvhfvsca.s | . . . 4 β’ Β· = ( Β·π βπ) | |
12 | 7, 8, 9, 10, 11 | dvhvsca 39567 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ β¨πΉ, πβ© β (π Γ πΈ))) β (π Β· β¨πΉ, πβ©) = β¨(π β(1st ββ¨πΉ, πβ©)), (π β (2nd ββ¨πΉ, πβ©))β©) |
13 | 1, 2, 6, 12 | syl12anc 836 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (π Β· β¨πΉ, πβ©) = β¨(π β(1st ββ¨πΉ, πβ©)), (π β (2nd ββ¨πΉ, πβ©))β©) |
14 | op1stg 7934 | . . . . 5 β’ ((πΉ β π β§ π β πΈ) β (1st ββ¨πΉ, πβ©) = πΉ) | |
15 | 3, 4, 14 | syl2anc 585 | . . . 4 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (1st ββ¨πΉ, πβ©) = πΉ) |
16 | 15 | fveq2d 6847 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (π β(1st ββ¨πΉ, πβ©)) = (π βπΉ)) |
17 | op2ndg 7935 | . . . . 5 β’ ((πΉ β π β§ π β πΈ) β (2nd ββ¨πΉ, πβ©) = π) | |
18 | 3, 4, 17 | syl2anc 585 | . . . 4 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (2nd ββ¨πΉ, πβ©) = π) |
19 | 18 | coeq2d 5819 | . . 3 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (π β (2nd ββ¨πΉ, πβ©)) = (π β π)) |
20 | 16, 19 | opeq12d 4839 | . 2 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β β¨(π β(1st ββ¨πΉ, πβ©)), (π β (2nd ββ¨πΉ, πβ©))β© = β¨(π βπΉ), (π β π)β©) |
21 | 13, 20 | eqtrd 2777 | 1 β’ (((πΎ β π β§ π β π») β§ (π β πΈ β§ πΉ β π β§ π β πΈ)) β (π Β· β¨πΉ, πβ©) = β¨(π βπΉ), (π β π)β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β¨cop 4593 Γ cxp 5632 β ccom 5638 βcfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 Β·π cvsca 17138 LHypclh 38450 LTrncltrn 38567 TEndoctendo 39218 DVecHcdvh 39544 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-sca 17150 df-vsca 17151 df-dvech 39545 |
This theorem is referenced by: dvhlveclem 39574 dib1dim2 39634 diclspsn 39660 dih1dimatlem 39795 |
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