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Theorem hhsssm 29669

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 2736	. . 3 ⊢ ( ·_𝑠OLD ‘𝑊) = ( ·_𝑠OLD ‘𝑊)
2	1	smfval 29016	. 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 6807	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
5		opex 5392	. . . . 5 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ V
6		normf 29534	. . . . . . 7 ⊢ norm_ℎ: ℋ⟶ℝ
7		ax-hilex 29410	. . . . . . 7 ⊢ ℋ ∈ V
8		fex 7134	. . . . . . 7 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → norm_ℎ ∈ V)
9	6, 7, 8	mp2an 690	. . . . . 6 ⊢ norm_ℎ ∈ V
10	9	resex 5951	. . . . 5 ⊢ (norm_ℎ ↾ 𝐻) ∈ V
11	5, 10	op1st 7871	. . . 4 ⊢ (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
12	4, 11	eqtri 2764	. . 3 ⊢ (1^st ‘𝑊) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
13	12	fveq2i 6807	. 2 ⊢ (2^nd ‘(1^st ‘𝑊)) = (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩)
14		hilablo 29571	. . . 4 ⊢ +_ℎ ∈ AbelOp
15		resexg 5949	. . . 4 ⊢ ( +_ℎ ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V)
16	14, 15	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V
17		hvmulex 29422	. . . 4 ⊢ ·_ℎ ∈ V
18	17	resex 5951	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) ∈ V
19	16, 18	op2nd 7872	. 2 ⊢ (2^nd ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩) = ( ·_ℎ ↾ (ℂ × 𝐻))
20	2, 13, 19	3eqtrri 2769	1 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) = ( ·_𝑠OLD ‘𝑊)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2104 Vcvv 3437 ⟨cop 4571 × cxp 5598 ↾ cres 5602 ⟶wf 6454 ‘cfv 6458 1^st c1st 7861 2^nd c2nd 7862 ℂcc 10919 ℝcr 10920 AbelOpcablo 28955 ·_𝑠OLD cns 28998 ℋchba 29330 +_ℎ cva 29331 ·_ℎ csm 29332 norm_ℎcno 29334

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978 ax-1cn 10979 ax-icn 10980 ax-addcl 10981 ax-addrcl 10982 ax-mulcl 10983 ax-mulrcl 10984 ax-mulcom 10985 ax-addass 10986 ax-mulass 10987 ax-distr 10988 ax-i2m1 10989 ax-1ne0 10990 ax-1rid 10991 ax-rnegex 10992 ax-rrecex 10993 ax-cnre 10994 ax-pre-lttri 10995 ax-pre-lttrn 10996 ax-pre-ltadd 10997 ax-pre-mulgt0 10998 ax-pre-sup 10999 ax-hilex 29410 ax-hfvadd 29411 ax-hvcom 29412 ax-hvass 29413 ax-hv0cl 29414 ax-hvaddid 29415 ax-hfvmul 29416 ax-hvmulid 29417 ax-hvdistr2 29420 ax-hvmul0 29421 ax-hfi 29490 ax-his1 29493 ax-his3 29495 ax-his4 29496

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3304 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-en 8765 df-dom 8766 df-sdom 8767 df-sup 9249 df-pnf 11061 df-mnf 11062 df-xr 11063 df-ltxr 11064 df-le 11065 df-sub 11257 df-neg 11258 df-div 11683 df-nn 12024 df-2 12086 df-3 12087 df-n0 12284 df-z 12370 df-uz 12633 df-rp 12781 df-seq 13772 df-exp 13833 df-cj 14859 df-re 14860 df-im 14861 df-sqrt 14995 df-grpo 28904 df-ablo 28956 df-sm 29008 df-hnorm 29379 df-hvsub 29382

This theorem is referenced by: hhsst 29677 hhsssh2 29681

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