kbmul - Hilbert Space Explorer

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Theorem kbmul 31984

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 31042	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		kbfval 31981	. . 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 1773	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))))
4		simp2 1136	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐵 ∈ ℋ)
5		cjcl 15141	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
6	5	3ad2ant1 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘𝐴) ∈ ℂ)
7		simp3 1137	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐶 ∈ ℋ)
8		hvmulcl 31042	. . . . 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 31981	. . . 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 1192	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐶 ∈ ℋ)
14		hicl 31109	. . . . . . 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 1190	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
17		simpl2 1191	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐵 ∈ ℋ)
18		ax-hvmulass 31036	. . . . . 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 1370	. . . . 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 11280	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝐴 · (𝑥 ·_ih 𝐶)))
21		his52 31116	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
22	16, 12, 13, 21	syl3anc 1370	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
23	20, 22	eqtr4d 2778	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)))
24	23	oveq1d 7446	. . . . 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 2777	. . . 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 5248	. . 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 2778	. 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 2778	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 1537 ∈ wcel 2106 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 · cmul 11158 ∗ccj 15132 ℋchba 30948 ·_ℎ csm 30950 ·_ih csp 30951 ketbra ck 30986

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hilex 31028 ax-hfvmul 31034 ax-hvmulass 31036 ax-hfi 31108 ax-his1 31111 ax-his3 31113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 df-kb 31880

This theorem is referenced by: kbass6 32150

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