| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvscacl | Structured version Visualization version GIF version | ||
| Description: Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-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 |
|---|---|
| dvhvscacl | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) ∈ (𝑇 × 𝐸)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvhfvsca.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 2 | dvhfvsca.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
| 3 | dvhfvsca.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
| 4 | dvhfvsca.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
| 5 | dvhfvsca.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
| 6 | 1, 2, 3, 4, 5 | dvhvsca 41081 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = 〈(𝑅‘(1st ‘𝐹)), (𝑅 ∘ (2nd ‘𝐹))〉) |
| 7 | simpl 482 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 8 | simprl 771 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → 𝑅 ∈ 𝐸) | |
| 9 | xp1st 8042 | . . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (1st ‘𝐹) ∈ 𝑇) | |
| 10 | 9 | ad2antll 729 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (1st ‘𝐹) ∈ 𝑇) |
| 11 | 1, 2, 3 | tendocl 40747 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐸 ∧ (1st ‘𝐹) ∈ 𝑇) → (𝑅‘(1st ‘𝐹)) ∈ 𝑇) |
| 12 | 7, 8, 10, 11 | syl3anc 1373 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅‘(1st ‘𝐹)) ∈ 𝑇) |
| 13 | xp2nd 8043 | . . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2nd ‘𝐹) ∈ 𝐸) | |
| 14 | 13 | ad2antll 729 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (2nd ‘𝐹) ∈ 𝐸) |
| 15 | 1, 3 | tendococl 40752 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐸 ∧ (2nd ‘𝐹) ∈ 𝐸) → (𝑅 ∘ (2nd ‘𝐹)) ∈ 𝐸) |
| 16 | 7, 8, 14, 15 | syl3anc 1373 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 ∘ (2nd ‘𝐹)) ∈ 𝐸) |
| 17 | opelxpi 5720 | . . 3 ⊢ (((𝑅‘(1st ‘𝐹)) ∈ 𝑇 ∧ (𝑅 ∘ (2nd ‘𝐹)) ∈ 𝐸) → 〈(𝑅‘(1st ‘𝐹)), (𝑅 ∘ (2nd ‘𝐹))〉 ∈ (𝑇 × 𝐸)) | |
| 18 | 12, 16, 17 | syl2anc 584 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → 〈(𝑅‘(1st ‘𝐹)), (𝑅 ∘ (2nd ‘𝐹))〉 ∈ (𝑇 × 𝐸)) |
| 19 | 6, 18 | eqeltrd 2840 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) ∈ (𝑇 × 𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4630 × cxp 5681 ∘ ccom 5687 ‘cfv 6559 (class class class)co 7429 1st c1st 8008 2nd c2nd 8009 ·𝑠 cvsca 17297 HLchlt 39329 LHypclh 39964 LTrncltrn 40081 TEndoctendo 40732 DVecHcdvh 41058 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-riotaBAD 38932 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-undef 8294 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-n0 12523 df-z 12610 df-uz 12875 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17244 df-plusg 17306 df-sca 17309 df-vsca 17310 df-proset 18336 df-poset 18355 df-plt 18371 df-lub 18387 df-glb 18388 df-join 18389 df-meet 18390 df-p0 18466 df-p1 18467 df-lat 18473 df-clat 18540 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39478 df-lplanes 39479 df-lvols 39480 df-lines 39481 df-psubsp 39483 df-pmap 39484 df-padd 39776 df-lhyp 39968 df-laut 39969 df-ldil 40084 df-ltrn 40085 df-trl 40139 df-tendo 40735 df-dvech 41059 |
| This theorem is referenced by: dvhlveclem 41088 diclspsn 41174 dih1dimatlem 41309 |
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