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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 485 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻)) | |
2 | simpr1 1190 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑅 ∈ 𝐸) | |
3 | simpr2 1191 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝐹 ∈ 𝑇) | |
4 | simpr3 1192 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑋 ∈ 𝐸) | |
5 | opelxpi 5594 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → 〈𝐹, 𝑋〉 ∈ (𝑇 × 𝐸)) | |
6 | 3, 4, 5 | syl2anc 586 | . . 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 38239 | . . 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 834 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑋〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉))〉) |
14 | op1stg 7703 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (1st ‘〈𝐹, 𝑋〉) = 𝐹) | |
15 | 3, 4, 14 | syl2anc 586 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑋〉) = 𝐹) |
16 | 15 | fveq2d 6676 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅‘(1st ‘〈𝐹, 𝑋〉)) = (𝑅‘𝐹)) |
17 | op2ndg 7704 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑋〉) = 𝑋) | |
18 | 3, 4, 17 | syl2anc 586 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑋〉) = 𝑋) |
19 | 18 | coeq2d 5735 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉)) = (𝑅 ∘ 𝑋)) |
20 | 16, 19 | opeq12d 4813 | . 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 2858 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 × cxp 5555 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 ·𝑠 cvsca 16571 LHypclh 37122 LTrncltrn 37239 TEndoctendo 37890 DVecHcdvh 38216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-sca 16583 df-vsca 16584 df-dvech 38217 |
This theorem is referenced by: dvhlveclem 38246 dib1dim2 38306 diclspsn 38332 dih1dimatlem 38467 |
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