normlem8 - Hilbert Space Explorer

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Theorem normlem8 31136

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 31109	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐴 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷)))
5	1, 2, 3, 4	mp3an 1463	. . 3 ⊢ (𝐴 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih 𝐶) + (𝐴 ·_ih 𝐷))
6		normlem8.2	. . . 4 ⊢ 𝐵 ∈ ℋ
7		his7 31109	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (𝐵 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷)))
8	6, 2, 3, 7	mp3an 1463	. . 3 ⊢ (𝐵 ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐵 ·_ih 𝐶) + (𝐵 ·_ih 𝐷))
9	5, 8	oveq12i 7443	. 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 31037	. . 3 ⊢ (𝐶 +_ℎ 𝐷) ∈ ℋ
11		ax-his2 31102	. . 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 1463	. 2 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐶 +_ℎ 𝐷)) = ((𝐴 ·_ih (𝐶 +_ℎ 𝐷)) + (𝐵 ·_ih (𝐶 +_ℎ 𝐷)))
13	1, 2	hicli 31100	. . 3 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
14	6, 3	hicli 31100	. . 3 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
15	1, 3	hicli 31100	. . 3 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
16	6, 2	hicli 31100	. . 3 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
17	13, 14, 15, 16	add42i 11487	. 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 2775	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 1540 ∈ wcel 2108 (class class class)co 7431 + caddc 11158 ℋchba 30938 +_ℎ cva 30939 ·_ih csp 30941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-hfvadd 31019 ax-hfi 31098 ax-his1 31101 ax-his2 31102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140

This theorem is referenced by: normlem9 31137 norm-ii-i 31156 normpythi 31161 normpari 31173 polid2i 31176

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