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Theorem hhssva 31416

Description: The vector addition 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
hhssva	⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) = ( +_𝑣 ‘𝑊)

**Proof of Theorem hhssva**
Step	Hyp	Ref	Expression
1		eqid 2761	. . 3 ⊢ ( +_𝑣 ‘𝑊) = ( +_𝑣 ‘𝑊)
2	1	vafval 30762	. 2 ⊢ ( +_𝑣 ‘𝑊) = (1^st ‘(1^st ‘𝑊))
3		hhss.1	. . . . 5 ⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
4	3	fveq2i 6864	. . . 4 ⊢ (1^st ‘𝑊) = (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩)
5		opex 5428	. . . . 5 ⊢ ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ ∈ V
6		normf 31282	. . . . . . 7 ⊢ norm_ℎ: ℋ⟶ℝ
7		ax-hilex 31158	. . . . . . 7 ⊢ ℋ ∈ V
8		fex 7204	. . . . . . 7 ⊢ ((norm_ℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → norm_ℎ ∈ V)
9	6, 7, 8	mp2an 702	. . . . . 6 ⊢ norm_ℎ ∈ V
10	9	resex 6011	. . . . 5 ⊢ (norm_ℎ ↾ 𝐻) ∈ V
11	5, 10	op1st 7972	. . . 4 ⊢ (1^st ‘⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
12	4, 11	eqtri 2784	. . 3 ⊢ (1^st ‘𝑊) = ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩
13	12	fveq2i 6864	. 2 ⊢ (1^st ‘(1^st ‘𝑊)) = (1^st ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩)
14		hilablo 31319	. . . 4 ⊢ +_ℎ ∈ AbelOp
15		resexg 6009	. . . 4 ⊢ ( +_ℎ ∈ AbelOp → ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V)
16	14, 15	ax-mp 5	. . 3 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) ∈ V
17		hvmulex 31170	. . . 4 ⊢ ·_ℎ ∈ V
18	17	resex 6011	. . 3 ⊢ ( ·_ℎ ↾ (ℂ × 𝐻)) ∈ V
19	16, 18	op1st 7972	. 2 ⊢ (1^st ‘⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩) = ( +_ℎ ↾ (𝐻 × 𝐻))
20	2, 13, 19	3eqtrri 2789	1 ⊢ ( +_ℎ ↾ (𝐻 × 𝐻)) = ( +_𝑣 ‘𝑊)

Colors of variables: wff setvar class

Syntax hints: = wceq 1559 ∈ wcel 2141 Vcvv 3453 ⟨cop 4585 × cxp 5641 ↾ cres 5645 ⟶wf 6511 ‘cfv 6515 1^st c1st 7962 ℂcc 11064 ℝcr 11065 AbelOpcablo 30703 +_𝑣 cpv 30744 ℋchba 31078 +_ℎ cva 31079 ·_ℎ csm 31080 norm_ℎcno 31082

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-hilex 31158 ax-hfvadd 31159 ax-hvcom 31160 ax-hvass 31161 ax-hv0cl 31162 ax-hvaddid 31163 ax-hfvmul 31164 ax-hvmulid 31165 ax-hvdistr2 31168 ax-hvmul0 31169 ax-hfi 31238 ax-his1 31241 ax-his3 31243 ax-his4 31244

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-er 8671 df-en 8921 df-dom 8922 df-sdom 8923 df-sup 9381 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-n0 12475 df-z 12562 df-uz 12833 df-rp 12987 df-seq 14008 df-exp 14068 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-grpo 30652 df-ablo 30704 df-va 30754 df-hnorm 31127 df-hvsub 31130

This theorem is referenced by: hhsst 31425 hhsssh2 31429

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