dvhopvsca - Mathbox for Norm Megill

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Theorem dvhopvsca 41679

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 486	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))
2		simpr1 1207	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑅 ∈ 𝐸)
3		simpr2 1208	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝐹 ∈ 𝑇)
4		simpr3 1209	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → 𝑋 ∈ 𝐸)
5		opelxpi 5682	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → ⟨𝐹, 𝑋⟩ ∈ (𝑇 × 𝐸))
6	3, 4, 5	syl2anc 593	. . 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 41678	. . 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 847	. 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 7976	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
15	3, 4, 14	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (1^st ‘⟨𝐹, 𝑋⟩) = 𝐹)
16	15	fveq2d 6865	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅‘(1^st ‘⟨𝐹, 𝑋⟩)) = (𝑅‘𝐹))
17		op2ndg 7977	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
18	3, 4, 17	syl2anc 593	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (2^nd ‘⟨𝐹, 𝑋⟩) = 𝑋)
19	18	coeq2d 5832	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 ∘ (2^nd ‘⟨𝐹, 𝑋⟩)) = (𝑅 ∘ 𝑋))
20	16, 19	opeq12d 4838	. 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 2796	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 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ⟨cop 4587 × cxp 5643 ∘ ccom 5649 ‘cfv 6515 (class class class)co 7390 1^st c1st 7962 2^nd c2nd 7963 ·_𝑠 cvsca 17271 LHypclh 40561 LTrncltrn 40678 TEndoctendo 41329 DVecHcdvh 41655

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-er 8671 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11213 df-mnf 11214 df-xr 11215 df-ltxr 11216 df-le 11217 df-sub 11411 df-neg 11412 df-nn 12206 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-n0 12477 df-z 12564 df-uz 12835 df-fz 13508 df-struct 17164 df-slot 17199 df-ndx 17211 df-base 17227 df-plusg 17280 df-sca 17283 df-vsca 17284 df-dvech 41656

This theorem is referenced by: dvhlveclem 41685 dib1dim2 41745 diclspsn 41771 dih1dimatlem 41906

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