normlem9 - Hilbert Space Explorer

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Theorem normlem9 29381

Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
normlem8.1	⊢ 𝐴 ∈ ℋ
normlem8.2	⊢ 𝐵 ∈ ℋ
normlem8.3	⊢ 𝐶 ∈ ℋ
normlem8.4	⊢ 𝐷 ∈ ℋ

**Assertion**
Ref	Expression
normlem9	⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) − ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)))

**Proof of Theorem normlem9**
Step	Hyp	Ref	Expression
1		normlem8.1	. . . 4 ⊢ 𝐴 ∈ ℋ
2		normlem8.2	. . . 4 ⊢ 𝐵 ∈ ℋ
3	1, 2	hvsubvali 29283	. . 3 ⊢ (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵))
4		normlem8.3	. . . 4 ⊢ 𝐶 ∈ ℋ
5		normlem8.4	. . . 4 ⊢ 𝐷 ∈ ℋ
6	4, 5	hvsubvali 29283	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7267	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
8		neg1cn 12017	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 29277	. . 3 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 29277	. . 3 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 29380	. 2 ⊢ ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = (((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) + ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)))
12		ax-his3 29347	. . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐷) ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷))))
13	8, 2, 10, 12	mp3an 1459	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷)))
14		his5 29349	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷)))
15	8, 2, 5, 14	mp3an 1459	. . . . . . 7 ⊢ (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷))
16	15	oveq2i 7266	. . . . . 6 ⊢ (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷))) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12018	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 14778	. . . . . . . . . . 11 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘-1) = -1
20	19	oveq2i 7266	. . . . . . . . 9 ⊢ (-1 · (∗‘-1)) = (-1 · -1)
21		ax-1cn 10860	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11353	. . . . . . . . 9 ⊢ (-1 · -1) = (1 · 1)
23	21	mulid2i 10911	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2770	. . . . . . . 8 ⊢ (-1 · (∗‘-1)) = 1
25	24	oveq1i 7265	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 14808	. . . . . . . 8 ⊢ (∗‘-1) ∈ ℂ
27	2, 5	hicli 29344	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 10917	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
29	27	mulid2i 10911	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2774	. . . . . 6 ⊢ (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2770	. . . . 5 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7266	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 29349	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷)))
34	8, 1, 5, 33	mp3an 1459	. . . . . . 7 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷))
35	19	oveq1i 7265	. . . . . . 7 ⊢ ((∗‘-1) · (𝐴 ·_ih 𝐷)) = (-1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 29344	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11350	. . . . . . 7 ⊢ (-1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2770	. . . . . 6 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 29347	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
40	8, 2, 4, 39	mp3an 1459	. . . . . . 7 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶))
41	2, 4	hicli 29344	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11350	. . . . . . 7 ⊢ (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2766	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7267	. . . . 5 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11235	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2769	. . . 4 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7267	. . 3 ⊢ (((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) + ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶))) = (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) + -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)))
48	1, 4	hicli 29344	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 10912	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 10912	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11229	. . 3 ⊢ (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) + -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))) = (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) − ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)))
52	47, 51	eqtri 2766	. 2 ⊢ (((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) + ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶))) = (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) − ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)))
53	7, 11, 52	3eqtri 2770	1 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) − ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 1c1 10803 + caddc 10805 · cmul 10807 − cmin 11135 -cneg 11136 ∗ccj 14735 ℋchba 29182 +_ℎ cva 29183 ·_ℎ csm 29184 ·_ih csp 29185 −_ℎ cmv 29188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-hfvadd 29263 ax-hfvmul 29268 ax-hfi 29342 ax-his1 29345 ax-his2 29346 ax-his3 29347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-2 11966 df-cj 14738 df-re 14739 df-im 14740 df-hvsub 29234

This theorem is referenced by: bcseqi 29383 normlem9at 29384 normpari 29417 polid2i 29420

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