dvhopvadd2 - Mathbox for Norm Megill

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Theorem dvhopvadd2 41562

Description: The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 41561 and/or dvhfplusr 41552. (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 41561	. 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 41552	. . . . 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 7381	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝑄(+_g‘(Scalar‘𝑈))𝑅) = (𝑄 + 𝑅))
13	12	opeq2d 4824	. 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 4574 ↦ cmpt 5167 ∘ ccom 5632 ‘cfv 6496 (class class class)co 7364 ∈ cmpo 7366 +_gcplusg 17217 Scalarcsca 17220 HLchlt 39818 LHypclh 40452 LTrncltrn 40569 TEndoctendo 41220 DVecHcdvh 41546

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

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 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-2 12241 df-3 12242 df-4 12243 df-5 12244 df-6 12245 df-n0 12435 df-z 12522 df-uz 12786 df-fz 13459 df-struct 17114 df-slot 17149 df-ndx 17161 df-base 17177 df-plusg 17230 df-mulr 17231 df-sca 17233 df-vsca 17234 df-edring 41225 df-dvech 41547

This theorem is referenced by: xihopellsmN 41722 dihopellsm 41723

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