his5 - Hilbert Space Explorer

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Theorem his5 30988

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 30915	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		ax-his1 30984	. . . . 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 1114	. . 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 30986	. . . 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 6844	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐴 ·_ℎ 𝐶) ·_ih 𝐵)) = (∗‘(𝐴 · (𝐶 ·_ih 𝐵))))
9		hicl 30982	. . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·_ih 𝐵) ∈ ℂ)
10		cjmul 15084	. . . . . 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 1114	. . . 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 30984	. . . . 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 7385	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) · (𝐵 ·_ih 𝐶)) = ((∗‘𝐴) · (∗‘(𝐶 ·_ih 𝐵))))
17	13, 16	eqtr4d 2767	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘(𝐴 · (𝐶 ·_ih 𝐵))) = ((∗‘𝐴) · (𝐵 ·_ih 𝐶)))
18	5, 8, 17	3eqtrd 2768	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 1086 = wceq 1540 ∈ wcel 2109 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 · cmul 11049 ∗ccj 15038 ℋchba 30821 ·_ℎ csm 30823 ·_ih csp 30824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-hfvmul 30907 ax-hfi 30981 ax-his1 30984 ax-his3 30986

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-cj 15041 df-re 15042 df-im 15043

This theorem is referenced by: his52 30989 his35 30990 normlem0 31011 normlem9 31020 bcseqi 31022 polid2i 31059 pjhthlem1 31293 eigrei 31736 eigposi 31738 eigorthi 31739 brafnmul 31853 lnopunilem1 31912 hmopm 31923 cnlnadjlem6 31974 adjlnop 31988

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