normlem9 - Hilbert Space Explorer

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Theorem normlem9 28889

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 28791	. . 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 28791	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7162	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
8		neg1cn 11745	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 28785	. . 3 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 28785	. . 3 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 28888	. 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 28855	. . . . . . 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 1457	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷)))
14		his5 28857	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷)))
15	8, 2, 5, 14	mp3an 1457	. . . . . . 7 ⊢ (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷))
16	15	oveq2i 7161	. . . . . 6 ⊢ (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷))) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 11746	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 14492	. . . . . . . . . . 11 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘-1) = -1
20	19	oveq2i 7161	. . . . . . . . 9 ⊢ (-1 · (∗‘-1)) = (-1 · -1)
21		ax-1cn 10589	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11082	. . . . . . . . 9 ⊢ (-1 · -1) = (1 · 1)
23	21	mulid2i 10640	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2848	. . . . . . . 8 ⊢ (-1 · (∗‘-1)) = 1
25	24	oveq1i 7160	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 14522	. . . . . . . 8 ⊢ (∗‘-1) ∈ ℂ
27	2, 5	hicli 28852	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 10646	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
29	27	mulid2i 10640	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2852	. . . . . 6 ⊢ (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2848	. . . . 5 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7161	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 28857	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷)))
34	8, 1, 5, 33	mp3an 1457	. . . . . . 7 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷))
35	19	oveq1i 7160	. . . . . . 7 ⊢ ((∗‘-1) · (𝐴 ·_ih 𝐷)) = (-1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 28852	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11079	. . . . . . 7 ⊢ (-1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2848	. . . . . 6 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 28855	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
40	8, 2, 4, 39	mp3an 1457	. . . . . . 7 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶))
41	2, 4	hicli 28852	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11079	. . . . . . 7 ⊢ (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2844	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7162	. . . . 5 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 10964	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2847	. . . 4 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7162	. . 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 28852	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 10641	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 10641	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 10958	. . 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 2844	. 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 2848	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 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 ℝcr 10530 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 -cneg 10865 ∗ccj 14449 ℋchba 28690 +_ℎ cva 28691 ·_ℎ csm 28692 ·_ih csp 28693 −_ℎ cmv 28696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-hfvadd 28771 ax-hfvmul 28776 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-2 11694 df-cj 14452 df-re 14453 df-im 14454 df-hvsub 28742

This theorem is referenced by: bcseqi 28891 normlem9at 28892 normpari 28925 polid2i 28928

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