kbmul - Hilbert Space Explorer

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Theorem kbmul 31891

Description: Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)

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

Proof of Theorem kbmul Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hvmulcl 30949	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		kbfval 31888	. . 3 ⊢ (((𝐴 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))))
3	1, 2	stoic3 1776	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))))
4		simp2 1137	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐵 ∈ ℋ)
5		cjcl 15078	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
6	5	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘𝐴) ∈ ℂ)
7		simp3 1138	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐶 ∈ ℋ)
8		hvmulcl 30949	. . . . 5 ⊢ (((∗‘𝐴) ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ)
9	6, 7, 8	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ)
10		kbfval 31888	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ) → (𝐵 ketbra ((∗‘𝐴) ·_ℎ 𝐶)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) ·_ℎ 𝐵)))
11	4, 9, 10	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ketbra ((∗‘𝐴) ·_ℎ 𝐶)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) ·_ℎ 𝐵)))
12		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ)
13		simpl3 1194	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐶 ∈ ℋ)
14		hicl 31016	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih 𝐶) ∈ ℂ)
15	12, 13, 14	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih 𝐶) ∈ ℂ)
16		simpl1 1192	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
17		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐵 ∈ ℋ)
18		ax-hvmulass 30943	. . . . . 6 ⊢ (((𝑥 ·_ih 𝐶) ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (((𝑥 ·_ih 𝐶) · 𝐴) ·_ℎ 𝐵) = ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵)))
19	15, 16, 17, 18	syl3anc 1373	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑥 ·_ih 𝐶) · 𝐴) ·_ℎ 𝐵) = ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵)))
20	15, 16	mulcomd 11202	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝐴 · (𝑥 ·_ih 𝐶)))
21		his52 31023	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
22	16, 12, 13, 21	syl3anc 1373	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
23	20, 22	eqtr4d 2768	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)))
24	23	oveq1d 7405	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑥 ·_ih 𝐶) · 𝐴) ·_ℎ 𝐵) = ((𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) ·_ℎ 𝐵))
25	19, 24	eqtr3d 2767	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵)) = ((𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) ·_ℎ 𝐵))
26	25	mpteq2dva 5203	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) ·_ℎ 𝐵)))
27	11, 26	eqtr4d 2768	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ketbra ((∗‘𝐴) ·_ℎ 𝐶)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))))
28	3, 27	eqtr4d 2768	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝐵 ketbra ((∗‘𝐴) ·_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 · cmul 11080 ∗ccj 15069 ℋchba 30855 ·_ℎ csm 30857 ·_ih csp 30858 ketbra ck 30893

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-hilex 30935 ax-hfvmul 30941 ax-hvmulass 30943 ax-hfi 31015 ax-his1 31018 ax-his3 31020

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-cj 15072 df-re 15073 df-im 15074 df-kb 31787

This theorem is referenced by: kbass6 32057

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