normlem8 - Hilbert Space Explorer

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Theorem normlem8 31098

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 31071	. . . 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 31071	. . . 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 7417	. 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 30999	. . 3 ⊢ (𝐶 +_ℎ 𝐷) ∈ ℋ
11		ax-his2 31064	. . 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 31062	. . 3 ⊢ (𝐴 ·_ih 𝐶) ∈ ℂ
14	6, 3	hicli 31062	. . 3 ⊢ (𝐵 ·_ih 𝐷) ∈ ℂ
15	1, 3	hicli 31062	. . 3 ⊢ (𝐴 ·_ih 𝐷) ∈ ℂ
16	6, 2	hicli 31062	. . 3 ⊢ (𝐵 ·_ih 𝐶) ∈ ℂ
17	13, 14, 15, 16	add42i 11461	. 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 2768	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 7405 + caddc 11132 ℋchba 30900 +_ℎ cva 30901 ·_ih csp 30903

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 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-hfvadd 30981 ax-hfi 31060 ax-his1 31063 ax-his2 31064

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-cj 15118 df-re 15119 df-im 15120

This theorem is referenced by: normlem9 31099 norm-ii-i 31118 normpythi 31123 normpari 31135 polid2i 31138

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