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 27327

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 27254	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		ax-his1 27323	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
3	1, 2	sylan2 490	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
4	3	3impb 1252	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
5	4	3com12 1261	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
6		ax-his3 27325	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
7	6	3com23 1263	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
8	7	fveq2d 6107	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·_ih 𝐵))))
9		hicl 27321	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·_ih 𝐵) ∈ ℂ)
10		cjmul 13730	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ·_ih 𝐵) ∈ ℂ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
11	9, 10	sylan2 490	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ)) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
12	11	3impb 1252	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
13	12	3com23 1263	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
14		ax-his1 27323	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
15	14	3adant1 1072	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
16	15	oveq2d 6565	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) · (𝐵 ·_ih 𝐶)) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
17	13, 16	eqtr4d 2647	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))
18	5, 8, 17	3eqtrd 2648	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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 · cmul 9820 ∗ccj 13684 ℋchil 27160 ·_ℎ csm 27162 ·_ih csp 27163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-hfvmul 27246 ax-hfi 27320 ax-his1 27323 ax-his3 27325

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689

This theorem is referenced by: his52 27328 his35 27329 normlem0 27350 normlem9 27359 bcseqi 27361 polid2i 27398 pjhthlem1 27634 eigrei 28077 eigposi 28079 eigorthi 28080 brafnmul 28194 lnopunilem1 28253 hmopm 28264 cnlnadjlem6 28315 adjlnop 28329

W3C validator