dvhvscacl - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvscacl		Structured version Visualization version GIF version

Theorem dvhvscacl 41053

Description: Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dvhvscacl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) ∈ (𝑇 × 𝐸))

**Proof of Theorem dvhvscacl**
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 41051	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩)
7		simpl 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simprl 770	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → 𝑅 ∈ 𝐸)
9		xp1st 8056	. . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (1^st ‘𝐹) ∈ 𝑇)
10	9	ad2antll 728	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (1^st ‘𝐹) ∈ 𝑇)
11	1, 2, 3	tendocl 40717	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐸 ∧ (1^st ‘𝐹) ∈ 𝑇) → (𝑅‘(1^st ‘𝐹)) ∈ 𝑇)
12	7, 8, 10, 11	syl3anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅‘(1^st ‘𝐹)) ∈ 𝑇)
13		xp2nd 8057	. . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
14	13	ad2antll 728	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (2^nd ‘𝐹) ∈ 𝐸)
15	1, 3	tendococl 40722	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐸 ∧ (2^nd ‘𝐹) ∈ 𝐸) → (𝑅 ∘ (2^nd ‘𝐹)) ∈ 𝐸)
16	7, 8, 14, 15	syl3anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 ∘ (2^nd ‘𝐹)) ∈ 𝐸)
17		opelxpi 5732	. . 3 ⊢ (((𝑅‘(1^st ‘𝐹)) ∈ 𝑇 ∧ (𝑅 ∘ (2^nd ‘𝐹)) ∈ 𝐸) → ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩ ∈ (𝑇 × 𝐸))
18	12, 16, 17	syl2anc 583	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩ ∈ (𝑇 × 𝐸))
19	6, 18	eqeltrd 2844	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 1537 ∈ wcel 2108 ⟨cop 4654 × cxp 5693 ∘ ccom 5699 ‘cfv 6568 (class class class)co 7443 1^st c1st 8022 2^nd c2nd 8023 ·_𝑠 cvsca 17309 HLchlt 39299 LHypclh 39934 LTrncltrn 40051 TEndoctendo 40702 DVecHcdvh 41028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7764 ax-cnex 11234 ax-resscn 11235 ax-1cn 11236 ax-icn 11237 ax-addcl 11238 ax-addrcl 11239 ax-mulcl 11240 ax-mulrcl 11241 ax-mulcom 11242 ax-addass 11243 ax-mulass 11244 ax-distr 11245 ax-i2m1 11246 ax-1ne0 11247 ax-1rid 11248 ax-rnegex 11249 ax-rrecex 11250 ax-cnre 11251 ax-pre-lttri 11252 ax-pre-lttrn 11253 ax-pre-ltadd 11254 ax-pre-mulgt0 11255 ax-riotaBAD 38902

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5650 df-we 5652 df-xp 5701 df-rel 5702 df-cnv 5703 df-co 5704 df-dm 5705 df-rn 5706 df-res 5707 df-ima 5708 df-pred 6327 df-ord 6393 df-on 6394 df-lim 6395 df-suc 6396 df-iota 6520 df-fun 6570 df-fn 6571 df-f 6572 df-f1 6573 df-fo 6574 df-f1o 6575 df-fv 6576 df-riota 7399 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7898 df-1st 8024 df-2nd 8025 df-undef 8308 df-frecs 8316 df-wrecs 8347 df-recs 8421 df-rdg 8460 df-1o 8516 df-er 8757 df-map 8880 df-en 8998 df-dom 8999 df-sdom 9000 df-fin 9001 df-pnf 11320 df-mnf 11321 df-xr 11322 df-ltxr 11323 df-le 11324 df-sub 11516 df-neg 11517 df-nn 12288 df-2 12350 df-3 12351 df-4 12352 df-5 12353 df-6 12354 df-n0 12548 df-z 12634 df-uz 12898 df-fz 13562 df-struct 17188 df-slot 17223 df-ndx 17235 df-base 17253 df-plusg 17318 df-sca 17321 df-vsca 17322 df-proset 18359 df-poset 18377 df-plt 18394 df-lub 18410 df-glb 18411 df-join 18412 df-meet 18413 df-p0 18489 df-p1 18490 df-lat 18496 df-clat 18563 df-oposet 39125 df-ol 39127 df-oml 39128 df-covers 39215 df-ats 39216 df-atl 39247 df-cvlat 39271 df-hlat 39300 df-llines 39448 df-lplanes 39449 df-lvols 39450 df-lines 39451 df-psubsp 39453 df-pmap 39454 df-padd 39746 df-lhyp 39938 df-laut 39939 df-ldil 40054 df-ltrn 40055 df-trl 40109 df-tendo 40705 df-dvech 41029

This theorem is referenced by: dvhlveclem 41058 diclspsn 41144 dih1dimatlem 41279

W3C validator