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Theorem kbmul 29734

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 28792	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		kbfval 29731	. . 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 1777	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) ketbra 𝐶) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih 𝐶) ·_ℎ (𝐴 ·_ℎ 𝐵))))
4		simp2 1133	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐵 ∈ ℋ)
5		cjcl 14466	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ)
6	5	3ad2ant1 1129	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘𝐴) ∈ ℂ)
7		simp3 1134	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐶 ∈ ℋ)
8		hvmulcl 28792	. . . . 5 ⊢ (((∗‘𝐴) ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ)
9	6, 7, 8	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((∗‘𝐴) ·_ℎ 𝐶) ∈ ℋ)
10		kbfval 29731	. . . 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 586	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ketbra ((∗‘𝐴) ·_ℎ 𝐶)) = (𝑥 ∈ ℋ ↦ ((𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) ·_ℎ 𝐵)))
12		simpr 487	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝑥 ∈ ℋ)
13		simpl3 1189	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐶 ∈ ℋ)
14		hicl 28859	. . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih 𝐶) ∈ ℂ)
15	12, 13, 14	syl2anc 586	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih 𝐶) ∈ ℂ)
16		simpl1 1187	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐴 ∈ ℂ)
17		simpl2 1188	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → 𝐵 ∈ ℋ)
18		ax-hvmulass 28786	. . . . . 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 1367	. . . . 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 10664	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝐴 · (𝑥 ·_ih 𝐶)))
21		his52 28866	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
22	16, 12, 13, 21	syl3anc 1367	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)) = (𝐴 · (𝑥 ·_ih 𝐶)))
23	20, 22	eqtr4d 2861	. . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝑥 ·_ih ((∗‘𝐴) ·_ℎ 𝐶)))
24	23	oveq1d 7173	. . . . 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 2860	. . . 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 5163	. . 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 2861	. 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 2861	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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 · cmul 10544 ∗ccj 14457 ℋchba 28698 ·_ℎ csm 28700 ·_ih csp 28701 ketbra ck 28736

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hilex 28778 ax-hfvmul 28784 ax-hvmulass 28786 ax-hfi 28858 ax-his1 28861 ax-his3 28863

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-kb 29630

This theorem is referenced by: kbass6 29900

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