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Theorem kbmul 32041

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 31099	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		kbfval 32038	. . 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 1778	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))))
4		simp2 1138	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐵 ∈ ℋ)
5		cjcl 15058	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
6	5	3ad2ant1 1134	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘𝐴) ∈ ℂ)
7		simp3 1139	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐶 ∈ ℋ)
8		hvmulcl 31099	. . . . 5 ⊢ (((∗‘𝐴) ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ)
9	6, 7, 8	syl2anc 585	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ)
10		kbfval 32038	. . . 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 585	. . 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 1195	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐶 ∈ ℋ)
14		hicl 31166	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih 𝐶) ∈ ℂ)
15	12, 13, 14	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih 𝐶) ∈ ℂ)
16		simpl1 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
17		simpl2 1194	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐵 ∈ ℋ)
18		ax-hvmulass 31093	. . . . . 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 1374	. . . . 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 11157	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝐴 · (𝑥 ·_ih 𝐶)))
21		his52 31173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
22	16, 12, 13, 21	syl3anc 1374	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
23	20, 22	eqtr4d 2775	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)))
24	23	oveq1d 7375	. . . . 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 2774	. . . 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 5179	. . 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 2775	. 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 2775	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 1087 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5167 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 · cmul 11034 ∗ccj 15049 ℋchba 31005 ·_ℎ csm 31007 ·_ih csp 31008 ketbra ck 31043

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-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-hilex 31085 ax-hfvmul 31091 ax-hvmulass 31093 ax-hfi 31165 ax-his1 31168 ax-his3 31170

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 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-cj 15052 df-re 15053 df-im 15054 df-kb 31937

This theorem is referenced by: kbass6 32207

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