normlem8 - Hilbert Space Explorer

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Theorem normlem8 31050

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
normlem8	⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐶 +_ℎ 𝐷)) = (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶)))

**Proof of Theorem normlem8**
Step	Hyp	Ref	Expression
1		normlem8.1	. . . 4 ⊢ 𝐴 ∈ ℋ
2		normlem8.3	. . . 4 ⊢ 𝐶 ∈ ℋ
3		normlem8.4	. . . 4 ⊢ 𝐷 ∈ ℋ
4		his7 31023	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷)))
5	1, 2, 3, 4	mp3an 1458	. . 3 ⊢ (𝐴 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷))
6		normlem8.2	. . . 4 ⊢ 𝐵 ∈ ℋ
7		his7 31023	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)))
8	6, 2, 3, 7	mp3an 1458	. . 3 ⊢ (𝐵 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
9	5, 8	oveq12i 7436	. 2 ⊢ ((𝐴 ·_ih (𝐶 +_ℎ 𝐷)) + (𝐵 ·_ih (𝐶 +_ℎ 𝐷))) = (((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷)) + ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)))
10	2, 3	hvaddcli 30951	. . 3 ⊢ (𝐶 +_ℎ 𝐷) ∈ ℋ
11		ax-his2 31016	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (𝐶 +_ℎ 𝐷) ∈ ℋ) → ((𝐴 +_ℎ 𝐵) ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih (𝐶 +_ℎ 𝐷)) + (𝐵 ·_ih (𝐶 +_ℎ 𝐷))))
12	1, 6, 10, 11	mp3an 1458	. 2 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih (𝐶 +_ℎ 𝐷)) + (𝐵 ·_ih (𝐶 +_ℎ 𝐷)))
13	1, 2	hicli 31014	. . 3 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
14	6, 3	hicli 31014	. . 3 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
15	1, 3	hicli 31014	. . 3 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
16	6, 2	hicli 31014	. . 3 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
17	13, 14, 15, 16	add42i 11489	. 2 ⊢ (((𝐴 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)) + ((𝐴 ·_ih 𝐷) + (𝐵 ·_ih 𝐶))) = (((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷)) + ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)))
18	9, 12, 17	3eqtr4i 2764	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

Syntax hints: = wceq 1534 ∈ wcel 2099 (class class class)co 7424 + caddc 11161 ℋchba 30852 +_ℎ cva 30853 ·_ih csp 30855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-hfvadd 30933 ax-hfi 31012 ax-his1 31015 ax-his2 31016

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-2 12327 df-cj 15104 df-re 15105 df-im 15106

This theorem is referenced by: normlem9 31051 norm-ii-i 31070 normpythi 31075 normpari 31087 polid2i 31090

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