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Mirrors > Home > HSE Home > Th. List > hhssva | Structured version Visualization version GIF version |
Description: The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhss.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
Ref | Expression |
---|---|
hhssva | ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
2 | 1 | vafval 30536 | . 2 ⊢ ( +𝑣 ‘𝑊) = (1st ‘(1st ‘𝑊)) |
3 | hhss.1 | . . . . 5 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
4 | 3 | fveq2i 6904 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
5 | opex 5470 | . . . . 5 ⊢ 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 ∈ V | |
6 | normf 31056 | . . . . . . 7 ⊢ normℎ: ℋ⟶ℝ | |
7 | ax-hilex 30932 | . . . . . . 7 ⊢ ℋ ∈ V | |
8 | fex 7243 | . . . . . . 7 ⊢ ((normℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → normℎ ∈ V) | |
9 | 6, 7, 8 | mp2an 690 | . . . . . 6 ⊢ normℎ ∈ V |
10 | 9 | resex 6038 | . . . . 5 ⊢ (normℎ ↾ 𝐻) ∈ V |
11 | 5, 10 | op1st 8011 | . . . 4 ⊢ (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
12 | 4, 11 | eqtri 2754 | . . 3 ⊢ (1st ‘𝑊) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
13 | 12 | fveq2i 6904 | . 2 ⊢ (1st ‘(1st ‘𝑊)) = (1st ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) |
14 | hilablo 31093 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
15 | resexg 6036 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V |
17 | hvmulex 30944 | . . . 4 ⊢ ·ℎ ∈ V | |
18 | 17 | resex 6038 | . . 3 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ∈ V |
19 | 16, 18 | op1st 8011 | . 2 ⊢ (1st ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) = ( +ℎ ↾ (𝐻 × 𝐻)) |
20 | 2, 13, 19 | 3eqtrri 2759 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 〈cop 4639 × cxp 5680 ↾ cres 5684 ⟶wf 6550 ‘cfv 6554 1st c1st 8001 ℂcc 11156 ℝcr 11157 AbelOpcablo 30477 +𝑣 cpv 30518 ℋchba 30852 +ℎ cva 30853 ·ℎ csm 30854 normℎcno 30856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-hilex 30932 ax-hfvadd 30933 ax-hvcom 30934 ax-hvass 30935 ax-hv0cl 30936 ax-hvaddid 30937 ax-hfvmul 30938 ax-hvmulid 30939 ax-hvdistr2 30942 ax-hvmul0 30943 ax-hfi 31012 ax-his1 31015 ax-his3 31017 ax-his4 31018 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-grpo 30426 df-ablo 30478 df-va 30528 df-hnorm 30901 df-hvsub 30904 |
This theorem is referenced by: hhsst 31199 hhsssh2 31203 |
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