dvhvaddcomN - Mathbox for Norm Megill

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Theorem dvhvaddcomN 41205

Description: Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
dvhvaddcomN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = (𝐺 + 𝐹))

Proof of Theorem dvhvaddcomN Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		xp1st 7953	. . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (1^st ‘𝐹) ∈ 𝑇)
3	2	ad2antrl 728	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (1^st ‘𝐹) ∈ 𝑇)
4		xp1st 7953	. . . . 5 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (1^st ‘𝐺) ∈ 𝑇)
5	4	ad2antll 729	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (1^st ‘𝐺) ∈ 𝑇)
6		dvhvaddcl.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		dvhvaddcl.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8	6, 7	ltrncom 40847	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1^st ‘𝐹) ∈ 𝑇 ∧ (1^st ‘𝐺) ∈ 𝑇) → ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))
9	1, 3, 5, 8	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((1^st ‘𝐹) ∘ (1^st ‘𝐺)) = ((1^st ‘𝐺) ∘ (1^st ‘𝐹)))
10		xp2nd 7954	. . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐹) ∈ 𝐸)
11		xp2nd 7954	. . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2^nd ‘𝐺) ∈ 𝐸)
12	10, 11	anim12i 613	. . . . 5 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸))
13		dvhvaddcl.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
14		eqid 2731	. . . . . . 7 ⊢ (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))
15	6, 7, 13, 14	tendoplcom 40891	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸) → ((2^nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐺)) = ((2^nd ‘𝐺)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐹)))
16	15	3expb 1120	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2^nd ‘𝐹) ∈ 𝐸 ∧ (2^nd ‘𝐺) ∈ 𝐸)) → ((2^nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐺)) = ((2^nd ‘𝐺)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐹)))
17	12, 16	sylan2 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2^nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐺)) = ((2^nd ‘𝐺)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐹)))
18		dvhvaddcl.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
19		dvhvaddcl.d	. . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑈)
20		dvhvaddcl.p	. . . . . . 7 ⊢ ⨣ = (+_g‘𝐷)
21	6, 7, 13, 18, 19, 14, 20	dvhfplusr 41193	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))))
22	21	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ⨣ = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))))
23	22	oveqd 7363	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) = ((2^nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐺)))
24	22	oveqd 7363	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐹)) = ((2^nd ‘𝐺)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2^nd ‘𝐹)))
25	17, 23, 24	3eqtr4d 2776	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)) = ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐹)))
26	9, 25	opeq12d 4830	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩ = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐹))⟩)
27		dvhvaddcl.a	. . 3 ⊢ + = (+_g‘𝑈)
28	6, 7, 13, 18, 19, 27, 20	dvhvadd 41201	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)
29	6, 7, 13, 18, 19, 27, 20	dvhvadd 41201	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐹) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐹))⟩)
30	29	ancom2s 650	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐺 + 𝐹) = ⟨((1^st ‘𝐺) ∘ (1^st ‘𝐹)), ((2^nd ‘𝐺) ⨣ (2^nd ‘𝐹))⟩)
31	26, 28, 30	3eqtr4d 2776	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = (𝐺 + 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⟨cop 4579 ↦ cmpt 5170 × cxp 5612 ∘ ccom 5618 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 +_gcplusg 17161 Scalarcsca 17164 HLchlt 39459 LHypclh 40093 LTrncltrn 40210 TEndoctendo 40861 DVecHcdvh 41187

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 39062

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-oposet 39285 df-ol 39287 df-oml 39288 df-covers 39375 df-ats 39376 df-atl 39407 df-cvlat 39431 df-hlat 39460 df-llines 39607 df-lplanes 39608 df-lvols 39609 df-lines 39610 df-psubsp 39612 df-pmap 39613 df-padd 39905 df-lhyp 40097 df-laut 40098 df-ldil 40213 df-ltrn 40214 df-trl 40268 df-tendo 40864 df-edring 40866 df-dvech 41188

This theorem is referenced by: (None)

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