dvhvsca - Mathbox for Norm Megill

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Theorem dvhvsca 41961

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)

**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
dvhvsca	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩)

Proof of Theorem dvhvsca Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhfvsca.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		dvhfvsca.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dvhfvsca.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		dvhfvsca.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		dvhfvsca.s	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑈)
6	1, 2, 3, 4, 5	dvhfvsca 41960	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩))
7	6	oveqd 7433	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑅 · 𝐹) = (𝑅(𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)𝐹))
8		eqid 2762	. . 3 ⊢ (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩) = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)
9	8	dvhvscaval 41959	. 2 ⊢ ((𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑅(𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)𝐹) = ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩)
10	7, 9	sylan9eq 2817	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = ⟨(𝑅‘(1^st ‘𝐹)), (𝑅 ∘ (2^nd ‘𝐹))⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4593 × cxp 5657 ∘ ccom 5663 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7987 2^nd c2nd 7988 ·_𝑠 cvsca 17350 LHypclh 40844 LTrncltrn 40961 TEndoctendo 41612 DVecHcdvh 41938

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13564 df-struct 17243 df-slot 17278 df-ndx 17290 df-base 17306 df-plusg 17359 df-sca 17362 df-vsca 17363 df-dvech 41939

This theorem is used by: dvhopvsca 41962 dvhvscacl 41963 dvhlveclem 41968 diblss 42030 dicvscacl 42051

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