his5 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > his5		Structured version Visualization version GIF version

Theorem his5 30377

Description: Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his5	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))

**Proof of Theorem his5**
Step	Hyp	Ref	Expression
1		hvmulcl 30304	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		ax-his1 30373	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
3	1, 2	sylan2 593	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
4	3	3impb 1115	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
5	4	3com12 1123	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
6		ax-his3 30375	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
7	6	3com23 1126	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
8	7	fveq2d 6895	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·_ih 𝐵))))
9		hicl 30371	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·_ih 𝐵) ∈ ℂ)
10		cjmul 15091	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ·_ih 𝐵) ∈ ℂ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
11	9, 10	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ)) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
12	11	3impb 1115	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
13	12	3com23 1126	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
14		ax-his1 30373	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
15	14	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
16	15	oveq2d 7427	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) · (𝐵 ·_ih 𝐶)) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
17	13, 16	eqtr4d 2775	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))
18	5, 8, 17	3eqtrd 2776	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 · cmul 11117 ∗ccj 15045 ℋchba 30210 ·_ℎ csm 30212 ·_ih csp 30213

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-hfvmul 30296 ax-hfi 30370 ax-his1 30373 ax-his3 30375

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-2 12277 df-cj 15048 df-re 15049 df-im 15050

This theorem is referenced by: his52 30378 his35 30379 normlem0 30400 normlem9 30409 bcseqi 30411 polid2i 30448 pjhthlem1 30682 eigrei 31125 eigposi 31127 eigorthi 31128 brafnmul 31242 lnopunilem1 31301 hmopm 31312 cnlnadjlem6 31363 adjlnop 31377

W3C validator