normlem8 - Hilbert Space Explorer

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Theorem normlem8 30625

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 30598	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷)))
5	1, 2, 3, 4	mp3an 1461	. . 3 ⊢ (𝐴 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷))
6		normlem8.2	. . . 4 ⊢ 𝐵 ∈ ℋ
7		his7 30598	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)))
8	6, 2, 3, 7	mp3an 1461	. . 3 ⊢ (𝐵 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
9	5, 8	oveq12i 7423	. 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 30526	. . 3 ⊢ (𝐶 +_ℎ 𝐷) ∈ ℋ
11		ax-his2 30591	. . 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 1461	. 2 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih (𝐶 +_ℎ 𝐷)) + (𝐵 ·_ih (𝐶 +_ℎ 𝐷)))
13	1, 2	hicli 30589	. . 3 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
14	6, 3	hicli 30589	. . 3 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
15	1, 3	hicli 30589	. . 3 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
16	6, 2	hicli 30589	. . 3 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
17	13, 14, 15, 16	add42i 11443	. 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 2770	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 1541 ∈ wcel 2106 (class class class)co 7411 + caddc 11115 ℋchba 30427 +_ℎ cva 30428 ·_ih csp 30430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-hfvadd 30508 ax-hfi 30587 ax-his1 30590 ax-his2 30591

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-2 12279 df-cj 15050 df-re 15051 df-im 15052

This theorem is referenced by: normlem9 30626 norm-ii-i 30645 normpythi 30650 normpari 30662 polid2i 30665

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