dvhvadd - Mathbox for Norm Megill

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Theorem dvhvadd 39085

Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)

**Hypotheses**
Ref	Expression
dvhvadd.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhvadd.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhvadd.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhvadd.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhvadd.f	⊢ 𝐷 = (Scalar‘𝑈)
dvhvadd.s	⊢ + = (+_g‘𝑈)
dvhvadd.p	⊢ ⨣ = (+_g‘𝐷)

**Assertion**
Ref	Expression
dvhvadd	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)

Proof of Theorem dvhvadd Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhvadd.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		dvhvadd.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dvhvadd.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		dvhvadd.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		dvhvadd.f	. . . 4 ⊢ 𝐷 = (Scalar‘𝑈)
6		dvhvadd.p	. . . 4 ⊢ ⨣ = (+_g‘𝐷)
7		eqid 2739	. . . 4 ⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩)
8		dvhvadd.s	. . . 4 ⊢ + = (+_g‘𝑈)
9	1, 2, 3, 4, 5, 6, 7, 8	dvhfvadd 39084	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩))
10	9	oveqd 7285	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 + 𝐺) = (𝐹(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩)𝐺))
11	7	dvhvaddval 39083	. 2 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩)𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)
12	10, 11	sylan9eq 2799	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ⟨cop 4572 × cxp 5586 ∘ ccom 5592 ‘cfv 6430 (class class class)co 7268 ∈ cmpo 7270 1^st c1st 7815 2^nd c2nd 7816 +_gcplusg 16943 Scalarcsca 16946 HLchlt 37343 LHypclh 37977 LTrncltrn 38094 TEndoctendo 38745 DVecHcdvh 39071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-edring 38750 df-dvech 39072

This theorem is referenced by: dvhopvadd 39086 dvhvaddcl 39088 dvhvaddcomN 39089 dvhvaddass 39090 dvhlveclem 39101 diblss 39163 dicvaddcl 39183

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