dvhopvadd2 - Mathbox for Norm Megill

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Theorem dvhopvadd2 41422

Description: The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 41421 and/or dvhfplusr 41412. (Contributed by NM, 26-Sep-2014.)

**Hypotheses**
Ref	Expression
dvhopvadd2.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhopvadd2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhopvadd2.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhopvadd2.p	⊢ + = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
dvhopvadd2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhopvadd2.s	⊢ ✚ = (+_g‘𝑈)

**Assertion**
Ref	Expression
dvhopvadd2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (⟨𝐹, 𝑄⟩ ✚ ⟨𝐺, 𝑅⟩) = ⟨(𝐹 ∘ 𝐺), (𝑄 + 𝑅)⟩)

Distinct variable groups: 𝑡,𝑠,𝐸 𝑓,𝐻 𝑓,𝑠,𝑡,𝐾 𝑓,𝑊,𝑠,𝑡 Allowed substitution hints: + (𝑡,𝑓,𝑠) ✚ (𝑡,𝑓,𝑠) 𝑄(𝑡,𝑓,𝑠) 𝑅(𝑡,𝑓,𝑠) 𝑇(𝑡,𝑓,𝑠) 𝑈(𝑡,𝑓,𝑠) 𝐸(𝑓) 𝐹(𝑡,𝑓,𝑠) 𝐺(𝑡,𝑓,𝑠) 𝐻(𝑡,𝑠)

**Proof of Theorem dvhopvadd2**
Step	Hyp	Ref	Expression
1		dvhopvadd2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		dvhopvadd2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dvhopvadd2.e	. . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		dvhopvadd2.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		eqid 2737	. . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
6		dvhopvadd2.s	. . 3 ⊢ ✚ = (+_g‘𝑈)
7		eqid 2737	. . 3 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
8	1, 2, 3, 4, 5, 6, 7	dvhopvadd 41421	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (⟨𝐹, 𝑄⟩ ✚ ⟨𝐺, 𝑅⟩) = ⟨(𝐹 ∘ 𝐺), (𝑄(+_g‘(Scalar‘𝑈))𝑅)⟩)
9		dvhopvadd2.p	. . . . . 6 ⊢ + = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
10	1, 2, 3, 4, 5, 9, 7	dvhfplusr 41412	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘(Scalar‘𝑈)) = + )
11	10	3ad2ant1 1134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (+_g‘(Scalar‘𝑈)) = + )
12	11	oveqd 7377	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝑄(+_g‘(Scalar‘𝑈))𝑅) = (𝑄 + 𝑅))
13	12	opeq2d 4837	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ⟨(𝐹 ∘ 𝐺), (𝑄(+_g‘(Scalar‘𝑈))𝑅)⟩ = ⟨(𝐹 ∘ 𝐺), (𝑄 + 𝑅)⟩)
14	8, 13	eqtrd 2772	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (⟨𝐹, 𝑄⟩ ✚ ⟨𝐺, 𝑅⟩) = ⟨(𝐹 ∘ 𝐺), (𝑄 + 𝑅)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⟨cop 4587 ↦ cmpt 5180 ∘ ccom 5629 ‘cfv 6493 (class class class)co 7360 ∈ cmpo 7362 +_gcplusg 17181 Scalarcsca 17184 HLchlt 39678 LHypclh 40312 LTrncltrn 40429 TEndoctendo 41080 DVecHcdvh 41406

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

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 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-struct 17078 df-slot 17113 df-ndx 17125 df-base 17141 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-edring 41085 df-dvech 41407

This theorem is referenced by: xihopellsmN 41582 dihopellsm 41583

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