his5 - Hilbert Space Explorer

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Theorem his5 31175

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 31102	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		ax-his1 31171	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
3	1, 2	sylan2 594	. . . 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 1124	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih (𝐴 ·_ℎ 𝐶)) = (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)))
6		ax-his3 31173	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
7	6	3com23 1127	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐶) ·_ih 𝐵) = (𝐴 · (𝐶 ·_ih 𝐵)))
8	7	fveq2d 6839	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·_ih 𝐵))))
9		hicl 31169	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·_ih 𝐵) ∈ ℂ)
10		cjmul 15098	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ·_ih 𝐵) ∈ ℂ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
11	9, 10	sylan2 594	. . . . 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 1127	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
14		ax-his1 31171	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
15	14	3adant1 1131	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) = (∗‘(𝐶 ·_ih 𝐵)))
16	15	oveq2d 7377	. . 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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6493 (class class class)co 7361 ℂcc 11030 · cmul 11037 ∗ccj 15052 ℋchba 31008 ·_ℎ csm 31010 ·_ih csp 31011

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-hfvmul 31094 ax-hfi 31168 ax-his1 31171 ax-his3 31173

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-cj 15055 df-re 15056 df-im 15057

This theorem is referenced by: his52 31176 his35 31177 normlem0 31198 normlem9 31207 bcseqi 31209 polid2i 31246 pjhthlem1 31480 eigrei 31923 eigposi 31925 eigorthi 31926 brafnmul 32040 lnopunilem1 32099 hmopm 32110 cnlnadjlem6 32161 adjlnop 32175

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