normlem9 - Hilbert Space Explorer

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Theorem normlem9 31602

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 31504	. . 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 31504	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7426	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
8		neg1cn 12230	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 31498	. . 3 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 31498	. . 3 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 31601	. 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 31568	. . . . . . 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 31570	. . . . . . . 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 7425	. . . . . 6 ⊢ (-1 · (𝐵 ·_ih (-1 ·_ℎ 𝐷))) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12231	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 15229	. . . . . . . . . . 11 ⊢ (-1 ∈ ℝ → (∗‘-1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘-1) = -1
20	19	oveq2i 7425	. . . . . . . . 9 ⊢ (-1 · (∗‘-1)) = (-1 · -1)
21		ax-1cn 11185	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11689	. . . . . . . . 9 ⊢ (-1 · -1) = (1 · 1)
23	21	mullidi 11241	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2787	. . . . . . . 8 ⊢ (-1 · (∗‘-1)) = 1
25	24	oveq1i 7424	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 15259	. . . . . . . 8 ⊢ (∗‘-1) ∈ ℂ
27	2, 5	hicli 31565	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 11247	. . . . . . 7 ⊢ ((-1 · (∗‘-1)) · (𝐵 ·_ih 𝐷)) = (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷)))
29	27	mullidi 11241	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2791	. . . . . 6 ⊢ (-1 · ((∗‘-1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2787	. . . . 5 ⊢ ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7425	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + ((-1 ·_ℎ 𝐵) ·_ih (-1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 31570	. . . . . . . 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 7424	. . . . . . 7 ⊢ ((∗‘-1) · (𝐴 ·_ih 𝐷)) = (-1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 31565	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11686	. . . . . . 7 ⊢ (-1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2787	. . . . . 6 ⊢ (𝐴 ·_ih (-1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 31568	. . . . . . . 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 31565	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11686	. . . . . . 7 ⊢ (-1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2783	. . . . . 6 ⊢ ((-1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7426	. . . . 5 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11569	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = (-(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2786	. . . 4 ⊢ ((𝐴 ·_ih (-1 ·_ℎ 𝐷)) + ((-1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7426	. . 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 31565	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 11242	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 11242	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11563	. . 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 2783	. 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 2787	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 2145 ‘cfv 6533 (class class class)co 7414 ℂcc 11125 ℝcr 11126 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11468 -cneg 11469 ∗ccj 15186 ℋchba 31403 +_ℎ cva 31404 ·_ℎ csm 31405 ·_ih csp 31406 −_ℎ cmv 31409

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-hfvadd 31484 ax-hfvmul 31489 ax-hfi 31563 ax-his1 31566 ax-his2 31567 ax-his3 31568

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-cj 15189 df-re 15190 df-im 15191 df-hvsub 31455

This theorem is used by: bcseqi 31604 normlem9at 31605 normpari 31638 polid2i 31641

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