normlem8 - Hilbert Space Explorer

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Theorem normlem8 31149

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 31122	. . . 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 31122	. . . 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 7460	. 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 31050	. . 3 ⊢ (𝐶 +_ℎ 𝐷) ∈ ℋ
11		ax-his2 31115	. . 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 31113	. . 3 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
14	6, 3	hicli 31113	. . 3 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
15	1, 3	hicli 31113	. . 3 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
16	6, 2	hicli 31113	. . 3 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
17	13, 14, 15, 16	add42i 11515	. 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 2778	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 1537 ∈ wcel 2108 (class class class)co 7448 + caddc 11187 ℋchba 30951 +_ℎ cva 30952 ·_ih csp 30954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-hfvadd 31032 ax-hfi 31111 ax-his1 31114 ax-his2 31115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-2 12356 df-cj 15148 df-re 15149 df-im 15150

This theorem is referenced by: normlem9 31150 norm-ii-i 31169 normpythi 31174 normpari 31186 polid2i 31189

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