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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 | ⊢ 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 482 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻)) | |
2 | simpr1 1193 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑅 ∈ 𝐸) | |
3 | simpr2 1194 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝐹 ∈ 𝑇) | |
4 | simpr3 1195 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑋 ∈ 𝐸) | |
5 | opelxpi 5726 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → 〈𝐹, 𝑋〉 ∈ (𝑇 × 𝐸)) | |
6 | 3, 4, 5 | syl2anc 584 | . . 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 41084 | . . 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 837 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘(1st ‘〈𝐹, 𝑋〉)), (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉))〉) |
14 | op1stg 8025 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (1st ‘〈𝐹, 𝑋〉) = 𝐹) | |
15 | 3, 4, 14 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑋〉) = 𝐹) |
16 | 15 | fveq2d 6911 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅‘(1st ‘〈𝐹, 𝑋〉)) = (𝑅‘𝐹)) |
17 | op2ndg 8026 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑋〉) = 𝑋) | |
18 | 3, 4, 17 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑋〉) = 𝑋) |
19 | 18 | coeq2d 5876 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 ∘ (2nd ‘〈𝐹, 𝑋〉)) = (𝑅 ∘ 𝑋)) |
20 | 16, 19 | opeq12d 4886 | . 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 2775 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 ∘ ccom 5693 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ·𝑠 cvsca 17302 LHypclh 39967 LTrncltrn 40084 TEndoctendo 40735 DVecHcdvh 41061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-sca 17314 df-vsca 17315 df-dvech 41062 |
This theorem is referenced by: dvhlveclem 41091 dib1dim2 41151 diclspsn 41177 dih1dimatlem 41312 |
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