normlem9 - Hilbert Space Explorer

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Theorem normlem9 31732

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 31634	. . 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 31634	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ ( -1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7439	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ ( -1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ ( -1 ·_ℎ 𝐷)))
8		neg1cn 12315	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 31628	. . 3 ⊢ ( -1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 31628	. . 3 ⊢ ( -1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 31731	. 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 31698	. . . . . . 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 31700	. . . . . . . 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 7438	. . . . . 6 ⊢ ( -1 · (𝐵 ·_ih ( -1 ·_ℎ 𝐷))) = ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12316	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 15318	. . . . . . . . . . 11 ⊢ ( -1 ∈ ℝ → (∗‘ -1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘ -1) = -1
20	19	oveq2i 7438	. . . . . . . . 9 ⊢ ( -1 · (∗‘ -1)) = ( -1 · -1)
21		ax-1cn 11268	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11774	. . . . . . . . 9 ⊢ ( -1 · -1) = (1 · 1)
23	21	mullidi 11324	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2790	. . . . . . . 8 ⊢ ( -1 · (∗‘ -1)) = 1
25	24	oveq1i 7437	. . . . . . 7 ⊢ (( -1 · (∗‘ -1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 15348	. . . . . . . 8 ⊢ (∗‘ -1) ∈ ℂ
27	2, 5	hicli 31695	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 11330	. . . . . . 7 ⊢ (( -1 · (∗‘ -1)) · (𝐵 ·_ih 𝐷)) = ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
29	27	mullidi 11324	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2794	. . . . . 6 ⊢ ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2790	. . . . 5 ⊢ (( -1 ·_ℎ 𝐵) ·_ih ( -1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7438	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (( -1 ·_ℎ 𝐵) ·_ih ( -1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 31700	. . . . . . . 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 7437	. . . . . . 7 ⊢ ((∗‘ -1) · (𝐴 ·_ih 𝐷)) = ( -1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 31695	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11771	. . . . . . 7 ⊢ ( -1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2790	. . . . . 6 ⊢ (𝐴 ·_ih ( -1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 31698	. . . . . . . 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 31695	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11771	. . . . . . 7 ⊢ ( -1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2786	. . . . . 6 ⊢ (( -1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7439	. . . . 5 ⊢ ((𝐴 ·_ih ( -1 ·_ℎ 𝐷)) + (( -1 ·_ℎ 𝐵) ·_ih 𝐶)) = ( -(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11652	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = ( -(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2789	. . . 4 ⊢ ((𝐴 ·_ih ( -1 ·_ℎ 𝐷)) + (( -1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7439	. . 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 31695	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 11325	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 11325	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11646	. . 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 2786	. 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 2790	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 1569 ∈ wcel 2146 ‘cfv 6546 (class class class)co 7427 ℂcc 11208 ℝcr 11209 1c1 11211 + caddc 11213 · cmul 11215 − cmin 11551 -cneg 11552 ∗ccj 15275 ℋchba 31533 +_ℎ cva 31534 ·_ℎ csm 31535 ·_ih csp 31536 −_ℎ cmv 31539

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-hfvadd 31614 ax-hfvmul 31619 ax-hfi 31693 ax-his1 31696 ax-his2 31697 ax-his3 31698

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-er 8728 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-cj 15278 df-re 15279 df-im 15280 df-hvsub 31585

This theorem is used by: bcseqi 31734 normlem9at 31735 normpari 31768 polid2i 31771

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