dvhvscacl - Mathbox for Norm Megill

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Theorem dvhvscacl 39117

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 39115	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩)
7		simpl 483	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simprl 768	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → 𝑅 ∈ 𝐸)
9		xp1st 7863	. . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (1^st ‘𝐹) ∈ 𝑇)
10	9	ad2antll 726	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (1^st ‘𝐹) ∈ 𝑇)
11	1, 2, 3	tendocl 38781	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐸 ∧ (1^st ‘𝐹) ∈ 𝑇) → (𝑅‘(1^st ‘𝐹)) ∈ 𝑇)
12	7, 8, 10, 11	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅‘(1^st ‘𝐹)) ∈ 𝑇)
13		xp2nd 7864	. . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
14	13	ad2antll 726	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (2^nd ‘𝐹) ∈ 𝐸)
15	1, 3	tendococl 38786	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑅 ∈ 𝐸 ∧ (2^nd ‘𝐹) ∈ 𝐸) → (𝑅 ∘ (2^nd ‘𝐹)) ∈ 𝐸)
16	7, 8, 14, 15	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 ∘ (2^nd ‘𝐹)) ∈ 𝐸)
17		opelxpi 5626	. . 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 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩ ∈ (𝑇 × 𝐸))
19	6, 18	eqeltrd 2839	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 396 = wceq 1539 ∈ wcel 2106 ⟨cop 4567 × cxp 5587 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 ·_𝑠 cvsca 16966 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 TEndoctendo 38766 DVecHcdvh 39092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-sca 16978 df-vsca 16979 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-dvech 39093

This theorem is referenced by: dvhlveclem 39122 diclspsn 39208 dih1dimatlem 39343

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