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Theorem kbmul 32043

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 31101	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
2		kbfval 32040	. . 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 15040	. . . . . 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 31101	. . . . 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 32040	. . . 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 31168	. . . . . . 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 31095	. . . . . 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 11165	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝐶) · 𝐴) = (𝐴 · (𝑥 ·_ih 𝐶)))
21		his52 31175	. . . . . . . 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 7383	. . . . 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 5193	. . 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 5181 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 · cmul 11043 ∗ccj 15031 ℋchba 31007 ·_ℎ csm 31009 ·_ih csp 31010 ketbra ck 31045

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-hilex 31087 ax-hfvmul 31093 ax-hvmulass 31095 ax-hfi 31167 ax-his1 31170 ax-his3 31172

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-cj 15034 df-re 15035 df-im 15036 df-kb 31939

This theorem is referenced by: kbass6 32209

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