normlem9 - Hilbert Space Explorer

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Theorem normlem9 31777

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 31679	. . 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 31679	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ ( -1 ·_ℎ 𝐷))
7	3, 6	oveq12i 7463	. 2 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ ( -1 ·_ℎ 𝐵)) ·_ih (𝐶 +_ℎ ( -1 ·_ℎ 𝐷)))
8		neg1cn 12345	. . . 4 ⊢ -1 ∈ ℂ
9	8, 2	hvmulcli 31673	. . 3 ⊢ ( -1 ·_ℎ 𝐵) ∈ ℋ
10	8, 5	hvmulcli 31673	. . 3 ⊢ ( -1 ·_ℎ 𝐷) ∈ ℋ
11	1, 9, 4, 10	normlem8 31776	. 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 31743	. . . . . . 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 1491	. . . . . 6 ⊢ (( -1 ·_ℎ 𝐵) ·_ih ( -1 ·_ℎ 𝐷)) = ( -1 · (𝐵 ·_ih ( -1 ·_ℎ 𝐷)))
14		his5 31745	. . . . . . . 8 ⊢ (( -1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih ( -1 ·_ℎ 𝐷)) = ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
15	8, 2, 5, 14	mp3an 1491	. . . . . . 7 ⊢ (𝐵 ·_ih ( -1 ·_ℎ 𝐷)) = ((∗‘ -1) · (𝐵 ·_ih 𝐷))
16	15	oveq2i 7462	. . . . . 6 ⊢ ( -1 · (𝐵 ·_ih ( -1 ·_ℎ 𝐷))) = ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
17		neg1rr 12346	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
18		cjre 15349	. . . . . . . . . . 11 ⊢ ( -1 ∈ ℝ → (∗‘ -1) = -1)
19	17, 18	ax-mp 5	. . . . . . . . . 10 ⊢ (∗‘ -1) = -1
20	19	oveq2i 7462	. . . . . . . . 9 ⊢ ( -1 · (∗‘ -1)) = ( -1 · -1)
21		ax-1cn 11297	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
22	21, 21	mul2negi 11803	. . . . . . . . 9 ⊢ ( -1 · -1) = (1 · 1)
23	21	mullidi 11353	. . . . . . . . 9 ⊢ (1 · 1) = 1
24	20, 22, 23	3eqtri 2790	. . . . . . . 8 ⊢ ( -1 · (∗‘ -1)) = 1
25	24	oveq1i 7461	. . . . . . 7 ⊢ (( -1 · (∗‘ -1)) · (𝐵 ·_ih 𝐷)) = (1 · (𝐵 ·_ih 𝐷))
26	8	cjcli 15379	. . . . . . . 8 ⊢ (∗‘ -1) ∈ ℂ
27	2, 5	hicli 31740	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
28	8, 26, 27	mulassi 11359	. . . . . . 7 ⊢ (( -1 · (∗‘ -1)) · (𝐵 ·_ih 𝐷)) = ( -1 · ((∗‘ -1) · (𝐵 ·_ih 𝐷)))
29	27	mullidi 11353	. . . . . . 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 7462	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (( -1 ·_ℎ 𝐵) ·_ih ( -1 ·_ℎ 𝐷))) = ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
33		his5 31745	. . . . . . . 8 ⊢ (( -1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih ( -1 ·_ℎ 𝐷)) = ((∗‘ -1) · (𝐴 ·_ih 𝐷)))
34	8, 1, 5, 33	mp3an 1491	. . . . . . 7 ⊢ (𝐴 ·_ih ( -1 ·_ℎ 𝐷)) = ((∗‘ -1) · (𝐴 ·_ih 𝐷))
35	19	oveq1i 7461	. . . . . . 7 ⊢ ((∗‘ -1) · (𝐴 ·_ih 𝐷)) = ( -1 · (𝐴 ·_ih 𝐷))
36	1, 5	hicli 31740	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
37	36	mulm1i 11800	. . . . . . 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 31743	. . . . . . . 8 ⊢ (( -1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (( -1 ·_ℎ 𝐵) ·_ih 𝐶) = ( -1 · (𝐵 ·_ih 𝐶)))
40	8, 2, 4, 39	mp3an 1491	. . . . . . 7 ⊢ (( -1 ·_ℎ 𝐵) ·_ih 𝐶) = ( -1 · (𝐵 ·_ih 𝐶))
41	2, 4	hicli 31740	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
42	41	mulm1i 11800	. . . . . . 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 7463	. . . . 5 ⊢ ((𝐴 ·_ih ( -1 ·_ℎ 𝐷)) + (( -1 ·_ℎ 𝐵) ·_ih 𝐶)) = ( -(𝐴 ·_ih 𝐷) + -(𝐵 ·_ih 𝐶))
45	36, 41	negdii 11681	. . . . 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 7463	. . 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 31740	. . . . 5 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
49	48, 27	addcli 11354	. . . 4 ⊢ ((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) ∈ ℂ
50	36, 41	addcli 11354	. . . 4 ⊢ ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)) ∈ ℂ
51	49, 50	negsubi 11675	. . 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 1570 ∈ wcel 2147 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 1c1 11240 + caddc 11242 · cmul 11244 − cmin 11580 -cneg 11581 ∗ccj 15306 ℋchba 31578 +_ℎ cva 31579 ·_ℎ csm 31580 ·_ih csp 31581 −_ℎ cmv 31584

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-hfvadd 31659 ax-hfvmul 31664 ax-hfi 31738 ax-his1 31741 ax-his2 31742 ax-his3 31743

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-cj 15309 df-re 15310 df-im 15311 df-hvsub 31630

This theorem is used by: bcseqi 31779 normlem9at 31780 normpari 31813 polid2i 31816

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