normlem9 - Hilbert Space Explorer

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Theorem normlem9 31543

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 31445	. . 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 31445	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7431	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
8		neg1cn 12220	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 31439	. . 3 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 31439	. . 3 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 31542	. 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 31509	. . . . . . 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 1490	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷)))
14		his5 31511	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷)))
15	8, 2, 5, 14	mp3an 1490	. . . . . . 7 ⊢ (𝐵 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐵 ·_ih 𝐷))
16	15	oveq2i 7430	. . . . . 6 ⊢ (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷))) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12221	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 15216	. . . . . . . . . . 11 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘-1) = -1
20	19	oveq2i 7430	. . . . . . . . 9 ⊢ (-1 · (∗‘-1)) = (-1 · -1)
21		ax-1cn 11175	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11679	. . . . . . . . 9 ⊢ (-1 · -1) = (1 · 1)
23	21	mullidi 11231	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2792	. . . . . . . 8 ⊢ (-1 · (∗‘-1)) = 1
25	24	oveq1i 7429	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 15246	. . . . . . . 8 ⊢ (∗‘-1) ∈ ℂ
27	2, 5	hicli 31506	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 11237	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
29	27	mullidi 11231	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2796	. . . . . 6 ⊢ (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2792	. . . . 5 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7430	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 31511	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷)))
34	8, 1, 5, 33	mp3an 1490	. . . . . . 7 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = ((∗‘-1) · (𝐴 ·_ih 𝐷))
35	19	oveq1i 7429	. . . . . . 7 ⊢ ((∗‘-1) · (𝐴 ·_ih 𝐷)) = (-1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 31506	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11676	. . . . . . 7 ⊢ (-1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2792	. . . . . 6 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 31509	. . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶)))
40	8, 2, 4, 39	mp3an 1490	. . . . . . 7 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = (-1 · (𝐵 ·_ih 𝐶))
41	2, 4	hicli 31506	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11676	. . . . . . 7 ⊢ (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2788	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7431	. . . . 5 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11559	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2791	. . . 4 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7431	. . 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 31506	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 11232	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 11232	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11553	. . 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 2788	. 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 2792	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

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2146 ‘cfv 6540 (class class class)co 7419 ℂcc 11115 ℝcr 11116 1c1 11118 + caddc 11120 · cmul 11122 − cmin 11458 -cneg 11459 ∗ccj 15173 ℋchba 31344 +_ℎ cva 31345 ·_ℎ csm 31346 ·_ih csp 31347 −_ℎ cmv 31350

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-hfvadd 31425 ax-hfvmul 31430 ax-hfi 31504 ax-his1 31507 ax-his2 31508 ax-his3 31509

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-cj 15176 df-re 15177 df-im 15178 df-hvsub 31396

This theorem is used by: bcseqi 31545 normlem9at 31546 normpari 31579 polid2i 31582

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