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Theorem hhsssm 31354

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 2740	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
2	1	smfval 30701	. 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 6837	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
5		opex 5410	. . . . 5 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ V
6		normf 31219	. . . . . . 7 ⊢ norm_ℎ: ℋ⟶ℝ
7		ax-hilex 31095	. . . . . . 7 ⊢ ℋ ∈ V
8		fex 7177	. . . . . . 7 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → norm_ℎ ∈ V)
9	6, 7, 8	mp2an 698	. . . . . 6 ⊢ norm_ℎ ∈ V
10	9	resex 5988	. . . . 5 ⊢ (norm_ℎ ↾ 𝐻) ∈ V
11	5, 10	op1st 7946	. . . 4 ⊢ (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
12	4, 11	eqtri 2763	. . 3 ⊢ (1^st ‘𝑊) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
13	12	fveq2i 6837	. 2 ⊢ (2^nd ‘(1^st ‘𝑊)) = (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩)
14		hilablo 31256	. . . 4 ⊢ +_ℎ ∈ AbelOp
15		resexg 5986	. . . 4 ⊢ ( +_ℎ ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V)
16	14, 15	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V
17		hvmulex 31107	. . . 4 ⊢ ·_ℎ ∈ V
18	17	resex 5988	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) ∈ V
19	16, 18	op2nd 7947	. 2 ⊢ (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩) = ( ·_ℎ ↾ (ℂ × 𝐻))
20	2, 13, 19	3eqtrri 2768	1 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) = ( ·_𝑠OLD ‘𝑊)

Colors of variables: wff setvar class

Syntax hints: = wceq 1547 ∈ wcel 2119 Vcvv 3432 ⟨cop 4568 × cxp 5623 ↾ cres 5627 ⟶wf 6488 ‘cfv 6492 1^st c1st 7936 2^nd c2nd 7937 ℂcc 11034 ℝcr 11035 AbelOpcablo 30640 ·_𝑠OLD cns 30683 ℋchba 31015 +_ℎ cva 31016 ·_ℎ csm 31017 norm_ℎcno 31019

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-hilex 31095 ax-hfvadd 31096 ax-hvcom 31097 ax-hvass 31098 ax-hv0cl 31099 ax-hvaddid 31100 ax-hfvmul 31101 ax-hvmulid 31102 ax-hvdistr2 31105 ax-hvmul0 31106 ax-hfi 31175 ax-his1 31178 ax-his3 31180 ax-his4 31181

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9352 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-n0 12436 df-z 12523 df-uz 12787 df-rp 12941 df-seq 13962 df-exp 14022 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-grpo 30589 df-ablo 30641 df-sm 30693 df-hnorm 31064 df-hvsub 31067

This theorem is referenced by: hhsst 31362 hhsssh2 31366

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