dvhopvsca - Mathbox for Norm Megill

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Theorem dvhopvsca 41467

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 482	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
2		simpr1 1196	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑅 ∈ 𝐸)
3		simpr2 1197	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝐹 ∈ 𝑇)
4		simpr3 1198	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑋 ∈ 𝐸)
5		opelxpi 5669	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → ⟨𝐹, 𝑋⟩ ∈ (𝑇 × 𝐸))
6	3, 4, 5	syl2anc 585	. . 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 41466	. . 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 837	. 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 7955	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
15	3, 4, 14	syl2anc 585	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
16	15	fveq2d 6846	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅‘(1^st ‘⟨𝐹, 𝑋⟩)) = (𝑅‘𝐹))
17		op2ndg 7956	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
18	3, 4, 17	syl2anc 585	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
19	18	coeq2d 5819	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 ∘ (2^nd ‘⟨𝐹, 𝑋⟩)) = (𝑅 ∘ 𝑋))
20	16, 19	opeq12d 4839	. 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 2772	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 × cxp 5630 ∘ ccom 5636 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 ·_𝑠 cvsca 17193 LHypclh 40349 LTrncltrn 40466 TEndoctendo 41117 DVecHcdvh 41443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-sca 17205 df-vsca 17206 df-dvech 41444

This theorem is referenced by: dvhlveclem 41473 dib1dim2 41533 diclspsn 41559 dih1dimatlem 41694

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