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Theorem hhsssm 28808

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 2772	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
2	1	smfval 28153	. 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 6496	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
5		opex 5207	. . . . 5 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ V
6		normf 28673	. . . . . . 7 ⊢ norm_ℎ: ℋ⟶ℝ
7		ax-hilex 28549	. . . . . . 7 ⊢ ℋ ∈ V
8		fex 6809	. . . . . . 7 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → norm_ℎ ∈ V)
9	6, 7, 8	mp2an 679	. . . . . 6 ⊢ norm_ℎ ∈ V
10	9	resex 5738	. . . . 5 ⊢ (norm_ℎ ↾ 𝐻) ∈ V
11	5, 10	op1st 7503	. . . 4 ⊢ (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
12	4, 11	eqtri 2796	. . 3 ⊢ (1^st ‘𝑊) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
13	12	fveq2i 6496	. 2 ⊢ (2^nd ‘(1^st ‘𝑊)) = (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩)
14		hilablo 28710	. . . 4 ⊢ +_ℎ ∈ AbelOp
15		resexg 5737	. . . 4 ⊢ ( +_ℎ ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V)
16	14, 15	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V
17		hvmulex 28561	. . . 4 ⊢ ·_ℎ ∈ V
18	17	resex 5738	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) ∈ V
19	16, 18	op2nd 7504	. 2 ⊢ (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩) = ( ·_ℎ ↾ (ℂ × 𝐻))
20	2, 13, 19	3eqtrri 2801	1 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) = ( ·_𝑠OLD ‘𝑊)

Colors of variables: wff setvar class

Syntax hints: = wceq 1507 ∈ wcel 2050 Vcvv 3409 ⟨cop 4441 × cxp 5399 ↾ cres 5403 ⟶wf 6178 ‘cfv 6182 1^st c1st 7493 2^nd c2nd 7494 ℂcc 10327 ℝcr 10328 AbelOpcablo 28092 ·_𝑠OLD cns 28135 ℋchba 28469 +_ℎ cva 28470 ·_ℎ csm 28471 norm_ℎcno 28473

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-hilex 28549 ax-hfvadd 28550 ax-hvcom 28551 ax-hvass 28552 ax-hv0cl 28553 ax-hvaddid 28554 ax-hfvmul 28555 ax-hvmulid 28556 ax-hvdistr2 28559 ax-hvmul0 28560 ax-hfi 28629 ax-his1 28632 ax-his3 28634 ax-his4 28635

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-sup 8695 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-n0 11702 df-z 11788 df-uz 12053 df-rp 12199 df-seq 13179 df-exp 13239 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-grpo 28041 df-ablo 28093 df-sm 28145 df-hnorm 28518 df-hvsub 28521

This theorem is referenced by: hhsst 28816 hhsssh2 28820

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