normlem9 - Hilbert Space Explorer

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Theorem normlem9 31142

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 31044	. . 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 31044	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7368	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
8		neg1cn 12128	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 31038	. . 3 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 31038	. . 3 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 31141	. 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 31108	. . . . . . 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 1463	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷)))
14		his5 31110	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷)))
15	8, 2, 5, 14	mp3an 1463	. . . . . . 7 ⊢ (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷))
16	15	oveq2i 7367	. . . . . 6 ⊢ (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷))) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12129	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 15060	. . . . . . . . . . 11 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘-1) = -1
20	19	oveq2i 7367	. . . . . . . . 9 ⊢ (-1 · (∗‘-1)) = (-1 · -1)
21		ax-1cn 11082	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11583	. . . . . . . . 9 ⊢ (-1 · -1) = (1 · 1)
23	21	mullidi 11135	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2761	. . . . . . . 8 ⊢ (-1 · (∗‘-1)) = 1
25	24	oveq1i 7366	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 15090	. . . . . . . 8 ⊢ (∗‘-1) ∈ ℂ
27	2, 5	hicli 31105	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 11141	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
29	27	mullidi 11135	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2765	. . . . . 6 ⊢ (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2761	. . . . 5 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7367	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 31110	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷)))
34	8, 1, 5, 33	mp3an 1463	. . . . . . 7 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷))
35	19	oveq1i 7366	. . . . . . 7 ⊢ ((∗‘-1) · (𝐴 ·_ih 𝐷)) = (-1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 31105	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11580	. . . . . . 7 ⊢ (-1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2761	. . . . . 6 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 31108	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
40	8, 2, 4, 39	mp3an 1463	. . . . . . 7 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶))
41	2, 4	hicli 31105	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11580	. . . . . . 7 ⊢ (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2757	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7368	. . . . 5 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11463	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2760	. . . 4 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7368	. . 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 31105	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 11136	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 11136	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11457	. . 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 2757	. 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 2761	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 1541 ∈ wcel 2113 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 1c1 11025 + caddc 11027 · cmul 11029 − cmin 11362 -cneg 11363 ∗ccj 15017 ℋchba 30943 +_ℎ cva 30944 ·_ℎ csm 30945 ·_ih csp 30946 −_ℎ cmv 30949

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-hfvadd 31024 ax-hfvmul 31029 ax-hfi 31103 ax-his1 31106 ax-his2 31107 ax-his3 31108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-cj 15020 df-re 15021 df-im 15022 df-hvsub 30995

This theorem is referenced by: bcseqi 31144 normlem9at 31145 normpari 31178 polid2i 31181

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