his2sub - Hilbert Space Explorer

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Theorem his2sub 31184

Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
his2sub	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))

**Proof of Theorem his2sub**
Step	Hyp	Ref	Expression
1		hvsubval 31108	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵)))
2	1	oveq1d 7383	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
3	2	3adant3 1133	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶))
4		neg1cn 12142	. . . . 5 ⊢ -1 ∈ ℂ
5		hvmulcl 31105	. . . . 5 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·_ℎ 𝐵) ∈ ℋ)
6	4, 5	mpan 691	. . . 4 ⊢ (𝐵 ∈ ℋ → (-1 ·_ℎ 𝐵) ∈ ℋ)
7		ax-his2 31175	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·_ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)))
8	6, 7	syl3an2 1165	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)))
9		ax-his3 31176	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
10	4, 9	mp3an1 1451	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
11		hicl 31172	. . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
12	11	mulm1d 11601	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶))
13	10, 12	eqtrd 2772	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶))
14	13	oveq2d 7384	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
15	14	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
16	8, 15	eqtrd 2772	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)))
17		hicl 31172	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
18	17	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ih 𝐶) ∈ ℂ)
19	11	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ih 𝐶) ∈ ℂ)
20	18, 19	negsubd 11510	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ih 𝐶) + -(𝐵 ·_ih 𝐶)) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))
21	3, 16, 20	3eqtrd 2776	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −_ℎ 𝐵) ·_ih 𝐶) = ((𝐴 ·_ih 𝐶) − (𝐵 ·_ih 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 (class class class)co 7368 ℂcc 11036 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11376 -cneg 11377 ℋchba 31011 +_ℎ cva 31012 ·_ℎ csm 31013 ·_ih csp 31014 −_ℎ cmv 31017

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-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-hfvmul 31097 ax-hfi 31171 ax-his2 31175 ax-his3 31176

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-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-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-id 5527 df-po 5540 df-so 5541 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-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-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-ltxr 11183 df-sub 11378 df-neg 11379 df-hvsub 31063

This theorem is referenced by: his2sub2 31185 hi2eq 31197 pjhthlem1 31483 h1de2i 31645 pjdifnormii 31775 lnopeqi 32100 riesz3i 32154 leop2 32216 hmopidmpji 32244 pjssposi 32264 pjclem4 32291 pj3si 32299

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