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Theorem normlem2 30993

Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
normlem1.1	⊢ 𝑆 ∈ ℂ
normlem1.2	⊢ 𝐹 ∈ ℋ
normlem1.3	⊢ 𝐺 ∈ ℋ
normlem2.4	⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))

**Assertion**
Ref	Expression
normlem2	⊢ 𝐵 ∈ ℝ

**Proof of Theorem normlem2**
Step	Hyp	Ref	Expression
1		normlem2.4	. 2 ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))
2		normlem1.1	. . . . . . . . 9 ⊢ 𝑆 ∈ ℂ
3	2	cjcli 15152	. . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ
4		normlem1.2	. . . . . . . . 9 ⊢ 𝐹 ∈ ℋ
5		normlem1.3	. . . . . . . . 9 ⊢ 𝐺 ∈ ℋ
6	4, 5	hicli 30963	. . . . . . . 8 ⊢ (𝐹 ·_ih 𝐺) ∈ ℂ
7	3, 6	mulcli 11253	. . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·_ih 𝐺)) ∈ ℂ
8	5, 4	hicli 30963	. . . . . . . 8 ⊢ (𝐺 ·_ih 𝐹) ∈ ℂ
9	2, 8	mulcli 11253	. . . . . . 7 ⊢ (𝑆 · (𝐺 ·_ih 𝐹)) ∈ ℂ
10	7, 9	cjaddi 15171	. . . . . 6 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))) = ((∗‘((∗‘𝑆) · (𝐹 ·_ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·_ih 𝐹))))
11	2	cjcji 15154	. . . . . . . . . 10 ⊢ (∗‘(∗‘𝑆)) = 𝑆
12	11	eqcomi 2734	. . . . . . . . 9 ⊢ 𝑆 = (∗‘(∗‘𝑆))
13	5, 4	his1i 30982	. . . . . . . . 9 ⊢ (𝐺 ·_ih 𝐹) = (∗‘(𝐹 ·_ih 𝐺))
14	12, 13	oveq12i 7431	. . . . . . . 8 ⊢ (𝑆 · (𝐺 ·_ih 𝐹)) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·_ih 𝐺)))
15	3, 6	cjmuli 15172	. . . . . . . 8 ⊢ (∗‘((∗‘𝑆) · (𝐹 ·_ih 𝐺))) = ((∗‘(∗‘𝑆)) · (∗‘(𝐹 ·_ih 𝐺)))
16	14, 15	eqtr4i 2756	. . . . . . 7 ⊢ (𝑆 · (𝐺 ·_ih 𝐹)) = (∗‘((∗‘𝑆) · (𝐹 ·_ih 𝐺)))
17	4, 5	his1i 30982	. . . . . . . . 9 ⊢ (𝐹 ·_ih 𝐺) = (∗‘(𝐺 ·_ih 𝐹))
18	17	oveq2i 7430	. . . . . . . 8 ⊢ ((∗‘𝑆) · (𝐹 ·_ih 𝐺)) = ((∗‘𝑆) · (∗‘(𝐺 ·_ih 𝐹)))
19	2, 8	cjmuli 15172	. . . . . . . 8 ⊢ (∗‘(𝑆 · (𝐺 ·_ih 𝐹))) = ((∗‘𝑆) · (∗‘(𝐺 ·_ih 𝐹)))
20	18, 19	eqtr4i 2756	. . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·_ih 𝐺)) = (∗‘(𝑆 · (𝐺 ·_ih 𝐹)))
21	16, 20	oveq12i 7431	. . . . . 6 ⊢ ((𝑆 · (𝐺 ·_ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·_ih 𝐺))) = ((∗‘((∗‘𝑆) · (𝐹 ·_ih 𝐺))) + (∗‘(𝑆 · (𝐺 ·_ih 𝐹))))
22	10, 21	eqtr4i 2756	. . . . 5 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))) = ((𝑆 · (𝐺 ·_ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·_ih 𝐺)))
23	7, 9	addcomi 11437	. . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹))) = ((𝑆 · (𝐺 ·_ih 𝐹)) + ((∗‘𝑆) · (𝐹 ·_ih 𝐺)))
24	22, 23	eqtr4i 2756	. . . 4 ⊢ (∗‘(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))) = (((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))
25	7, 9	addcli 11252	. . . . 5 ⊢ (((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹))) ∈ ℂ
26	25	cjrebi 15157	. . . 4 ⊢ ((((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹))) ∈ ℝ ↔ (∗‘(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹)))) = (((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹))))
27	24, 26	mpbir 230	. . 3 ⊢ (((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹))) ∈ ℝ
28	27	renegcli 11553	. 2 ⊢ -(((∗‘𝑆) · (𝐹 ·_ih 𝐺)) + (𝑆 · (𝐺 ·_ih 𝐹))) ∈ ℝ
29	1, 28	eqeltri 2821	1 ⊢ 𝐵 ∈ ℝ

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 + caddc 11143 · cmul 11145 -cneg 11477 ∗ccj 15079 ℋchba 30801 ·_ih csp 30804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-hfi 30961 ax-his1 30964

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-2 12308 df-cj 15082 df-re 15083 df-im 15084

This theorem is referenced by: normlem3 30994 normlem6 30997 normlem7 30998 norm-ii-i 31019

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