his2sub2 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > his2sub2		Structured version Visualization version GIF version

Theorem his2sub2 31688

Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his2sub2	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)))

**Proof of Theorem his2sub2**
Step	Hyp	Ref	Expression
1		his2sub 31687	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴)))
2	1	fveq2d 6887	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))))
3		hicl 31675	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·_ih 𝐴) ∈ ℂ)
4		hicl 31675	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·_ih 𝐴) ∈ ℂ)
5		cjsub 15309	. . . . . 6 ⊢ (((𝐵 ·_ih 𝐴) ∈ ℂ ∧ (𝐶 ·_ih 𝐴) ∈ ℂ) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
6	3, 4, 5	syl2an 608	. . . . 5 ⊢ (((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ)) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
7	6	3impdir 1370	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 ·_ih 𝐴) − (𝐶 ·_ih 𝐴))) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
8	2, 7	eqtrd 2796	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
9	8	3comr 1143	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
10		hvsubcl 31612	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		ax-his1 31677	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 −_ℎ 𝐶) ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
12	10, 11	sylan2 605	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
13	12	3impb 1132	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = (∗‘((𝐵 −_ℎ 𝐶) ·_ih 𝐴)))
14		ax-his1 31677	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
15	14	3adant3 1150	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐵) = (∗‘(𝐵 ·_ih 𝐴)))
16		ax-his1 31677	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
17	16	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐴)))
18	15, 17	oveq12d 7436	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)) = ((∗‘(𝐵 ·_ih 𝐴)) − (∗‘(𝐶 ·_ih 𝐴))))
19	9, 13, 18	3eqtr4d 2806	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ih 𝐵) − (𝐴 ·_ih 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 − cmin 11534 ∗ccj 15256 ℋchba 31514 ·_ih csp 31517 −_ℎ cmv 31520

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-hfvadd 31595 ax-hfvmul 31600 ax-hfi 31674 ax-his1 31677 ax-his2 31678 ax-his3 31679

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-cj 15259 df-re 15260 df-im 15261 df-hvsub 31566

This theorem is used by: pjhthlem1 31986 riesz4i 32658

W3C validator