normlem9 - Hilbert Space Explorer

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Theorem normlem9 31720

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 31622	. . 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 31622	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ ( -1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7432	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ ( -1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ ( -1 ·_ℎ 𝐷)))
8		neg1cn 12305	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 31616	. . 3 ⊢ ( -1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 31616	. . 3 ⊢ ( -1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 31719	. 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 31686	. . . . . . 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 31688	. . . . . . . 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 7431	. . . . . 6 ⊢ ( -1 · (𝐵 ·_ih ( -1 ·_ℎ 𝐷))) = ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12306	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 15306	. . . . . . . . . . 11 ⊢ ( -1 ∈ ℝ → (∗‘ -1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘ -1) = -1
20	19	oveq2i 7431	. . . . . . . . 9 ⊢ ( -1 · (∗‘ -1)) = ( -1 · -1)
21		ax-1cn 11258	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11764	. . . . . . . . 9 ⊢ ( -1 · -1) = (1 · 1)
23	21	mullidi 11314	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2788	. . . . . . . 8 ⊢ ( -1 · (∗‘ -1)) = 1
25	24	oveq1i 7430	. . . . . . 7 ⊢ (( -1 · (∗‘ -1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 15336	. . . . . . . 8 ⊢ (∗‘ -1) ∈ ℂ
27	2, 5	hicli 31683	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 11320	. . . . . . 7 ⊢ (( -1 · (∗‘ -1)) · (𝐵 ·_ih 𝐷)) = ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
29	27	mullidi 11314	. . . . . . 7 ⊢ (1 · (𝐵 ·_ih 𝐷)) = (𝐵 ·_ih 𝐷)
30	25, 28, 29	3eqtr3i 2792	. . . . . 6 ⊢ ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷))) = (𝐵 ·_ih 𝐷)
31	13, 16, 30	3eqtri 2788	. . . . 5 ⊢ (( -1 ·_ℎ 𝐵) ·_ih ( -1 ·_ℎ 𝐷)) = (𝐵 ·_ih 𝐷)
32	31	oveq2i 7431	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (( -1 ·_ℎ 𝐵) ·_ih ( -1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 31688	. . . . . . . 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 7430	. . . . . . 7 ⊢ ((∗‘ -1) · (𝐴 ·_ih 𝐷)) = ( -1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 31683	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11761	. . . . . . 7 ⊢ ( -1 · (𝐴 ·_ih 𝐷)) = -(𝐴 ·_ih 𝐷)
38	34, 35, 37	3eqtri 2788	. . . . . 6 ⊢ (𝐴 ·_ih ( -1 ·_ℎ 𝐷)) = -(𝐴 ·_ih 𝐷)
39		ax-his3 31686	. . . . . . . 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 31683	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11761	. . . . . . 7 ⊢ ( -1 · (𝐵 ·_ih 𝐶)) = -(𝐵 ·_ih 𝐶)
43	40, 42	eqtri 2784	. . . . . 6 ⊢ (( -1 ·_ℎ 𝐵) ·_ih 𝐶) = -(𝐵 ·_ih 𝐶)
44	38, 43	oveq12i 7432	. . . . 5 ⊢ ((𝐴 ·_ih ( -1 ·_ℎ 𝐷)) + (( -1 ·_ℎ 𝐵) ·_ih 𝐶)) = ( -(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11642	. . . . 5 ⊢ -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) = ( -(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
46	44, 45	eqtr4i 2787	. . . 4 ⊢ ((𝐴 ·_ih ( -1 ·_ℎ 𝐷)) + (( -1 ·_ℎ 𝐵) ·_ih 𝐶)) = -((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))
47	32, 46	oveq12i 7432	. . 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 31683	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 11315	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 11315	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11636	. . 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 2784	. 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 2788	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 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 1c1 11201 + caddc 11203 · cmul 11205 − cmin 11541 -cneg 11542 ∗ccj 15263 ℋchba 31521 +_ℎ cva 31522 ·_ℎ csm 31523 ·_ih csp 31524 −_ℎ cmv 31527

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 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-hfvadd 31602 ax-hfvmul 31607 ax-hfi 31681 ax-his1 31684 ax-his2 31685 ax-his3 31686

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-cj 15266 df-re 15267 df-im 15268 df-hvsub 31573

This theorem is used by: bcseqi 31722 normlem9at 31723 normpari 31756 polid2i 31759

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