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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 | ⊢ 𝐻 = (LHyp‘𝐾) |
dvhfvsca.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvhfvsca.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dvhfvsca.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvhfvsca.s | ⊢ · = ( ·𝑠 ‘𝑈) |
Ref | Expression |
---|---|
dvhopvsca | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑋)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻)) | |
2 | simpr1 1067 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑅 ∈ 𝐸) | |
3 | simpr2 1068 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝐹 ∈ 𝑇) | |
4 | simpr3 1069 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑋 ∈ 𝐸) | |
5 | opelxpi 5148 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → 〈𝐹, 𝑋〉 ∈ (𝑇 × 𝐸)) | |
6 | 3, 4, 5 | syl2anc 693 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 〈𝐹, 𝑋〉 ∈ (𝑇 × 𝐸)) |
7 | dvhfvsca.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | dvhfvsca.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
9 | dvhfvsca.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
10 | dvhfvsca.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
11 | dvhfvsca.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑈) | |
12 | 7, 8, 9, 10, 11 | dvhvsca 36390 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 〈𝐹, 𝑋〉 ∈ (𝑇 × 𝐸))) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑋〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉))〉) |
13 | 1, 2, 6, 12 | syl12anc 1324 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑋〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉))〉) |
14 | op1stg 7180 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (1st ‘〈𝐹, 𝑋〉) = 𝐹) | |
15 | 3, 4, 14 | syl2anc 693 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑋〉) = 𝐹) |
16 | 15 | fveq2d 6195 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅‘(1st ‘〈𝐹, 𝑋〉)) = (𝑅‘𝐹)) |
17 | op2ndg 7181 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑋〉) = 𝑋) | |
18 | 3, 4, 17 | syl2anc 693 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑋〉) = 𝑋) |
19 | 18 | coeq2d 5284 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉)) = (𝑅 ∘ 𝑋)) |
20 | 16, 19 | opeq12d 4410 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 〈(𝑅‘(1st ‘〈𝐹, 𝑋〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉))〉 = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑋)〉) |
21 | 13, 20 | eqtrd 2656 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑋)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 × cxp 5112 ∘ ccom 5118 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 ·𝑠 cvsca 15945 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 DVecHcdvh 36367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-sca 15957 df-vsca 15958 df-dvech 36368 |
This theorem is referenced by: dvhlveclem 36397 dib1dim2 36457 diclspsn 36483 dih1dimatlem 36618 |
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