dvhopvsca - Mathbox for Norm Megill

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Theorem dvhopvsca 41738

Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-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
dvhopvsca	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · ⟨𝐹, 𝑋⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑋)⟩)

**Proof of Theorem dvhopvsca**
Step	Hyp	Ref	Expression
1		simpl 487	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
2		simpr1 1211	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑅 ∈ 𝐸)
3		simpr2 1212	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝐹 ∈ 𝑇)
4		simpr3 1213	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑋 ∈ 𝐸)
5		opelxpi 5689	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → ⟨𝐹, 𝑋⟩ ∈ (𝑇 × 𝐸))
6	3, 4, 5	syl2anc 595	. . 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 41737	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ ⟨𝐹, 𝑋⟩ ∈ (𝑇 × 𝐸))) → (𝑅 · ⟨𝐹, 𝑋⟩) = ⟨(𝑅‘(1^st ‘⟨𝐹, 𝑋⟩)), (𝑅 ∘ (2^nd ‘⟨𝐹, 𝑋⟩))⟩)
13	1, 2, 6, 12	syl12anc 849	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · ⟨𝐹, 𝑋⟩) = ⟨(𝑅‘(1^st ‘⟨𝐹, 𝑋⟩)), (𝑅 ∘ (2^nd ‘⟨𝐹, 𝑋⟩))⟩)
14		op1stg 7986	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
15	3, 4, 14	syl2anc 595	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
16	15	fveq2d 6875	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅‘(1^st ‘⟨𝐹, 𝑋⟩)) = (𝑅‘𝐹))
17		op2ndg 7987	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
18	3, 4, 17	syl2anc 595	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
19	18	coeq2d 5839	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 ∘ (2^nd ‘⟨𝐹, 𝑋⟩)) = (𝑅 ∘ 𝑋))
20	16, 19	opeq12d 4842	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → ⟨(𝑅‘(1^st ‘⟨𝐹, 𝑋⟩)), (𝑅 ∘ (2^nd ‘⟨𝐹, 𝑋⟩))⟩ = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑋)⟩)
21	13, 20	eqtrd 2800	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 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ⟨cop 4591 × cxp 5650 ∘ ccom 5656 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 ·_𝑠 cvsca 17304 LHypclh 40620 LTrncltrn 40737 TEndoctendo 41388 DVecHcdvh 41714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-plusg 17313 df-sca 17316 df-vsca 17317 df-dvech 41715

This theorem is referenced by: dvhlveclem 41744 dib1dim2 41804 diclspsn 41830 dih1dimatlem 41965

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