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Theorem hhsssm 28115

Description: The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
hhss.1	⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩

**Assertion**
Ref	Expression
hhsssm	⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) = ( ·_𝑠OLD ‘𝑊)

**Proof of Theorem hhsssm**
Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
2	1	smfval 27460	. 2 ⊢ ( ·_𝑠OLD ‘𝑊) = (2^nd ‘(1^st ‘𝑊))
3		hhss.1	. . . . 5 ⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
4	3	fveq2i 6194	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
5		opex 4932	. . . . 5 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ V
6		normf 27980	. . . . . . 7 ⊢ norm_ℎ: ℋ⟶ℝ
7		ax-hilex 27856	. . . . . . 7 ⊢ ℋ ∈ V
8		fex 6490	. . . . . . 7 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → norm_ℎ ∈ V)
9	6, 7, 8	mp2an 708	. . . . . 6 ⊢ norm_ℎ ∈ V
10	9	resex 5443	. . . . 5 ⊢ (norm_ℎ ↾ 𝐻) ∈ V
11	5, 10	op1st 7176	. . . 4 ⊢ (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
12	4, 11	eqtri 2644	. . 3 ⊢ (1^st ‘𝑊) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
13	12	fveq2i 6194	. 2 ⊢ (2^nd ‘(1^st ‘𝑊)) = (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩)
14		hilablo 28017	. . . 4 ⊢ +_ℎ ∈ AbelOp
15		resexg 5442	. . . 4 ⊢ ( +_ℎ ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V)
16	14, 15	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V
17		hvmulex 27868	. . . 4 ⊢ ·_ℎ ∈ V
18	17	resex 5443	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) ∈ V
19	16, 18	op2nd 7177	. 2 ⊢ (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩) = ( ·_ℎ ↾ (ℂ × 𝐻))
20	2, 13, 19	3eqtrri 2649	1 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) = ( ·_𝑠OLD ‘𝑊)

Colors of variables: wff setvar class

Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⟨cop 4183 × cxp 5112 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 1^st c1st 7166 2^nd c2nd 7167 ℂcc 9934 ℝcr 9935 AbelOpcablo 27398 ·_𝑠OLD cns 27442 ℋchil 27776 +_ℎ cva 27777 ·_ℎ csm 27778 norm_ℎcno 27780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his3 27941 ax-his4 27942

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-grpo 27347 df-ablo 27399 df-sm 27452 df-hnorm 27825 df-hvsub 27828

This theorem is referenced by: hhsst 28123 hhsssh2 28127

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