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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 2735 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
2 | 1 | vafval 30632 | . 2 ⊢ ( +𝑣 ‘𝑊) = (1st ‘(1st ‘𝑊)) |
3 | hhss.1 | . . . . 5 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
4 | 3 | fveq2i 6910 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
5 | opex 5475 | . . . . 5 ⊢ 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 ∈ V | |
6 | normf 31152 | . . . . . . 7 ⊢ normℎ: ℋ⟶ℝ | |
7 | ax-hilex 31028 | . . . . . . 7 ⊢ ℋ ∈ V | |
8 | fex 7246 | . . . . . . 7 ⊢ ((normℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → normℎ ∈ V) | |
9 | 6, 7, 8 | mp2an 692 | . . . . . 6 ⊢ normℎ ∈ V |
10 | 9 | resex 6049 | . . . . 5 ⊢ (normℎ ↾ 𝐻) ∈ V |
11 | 5, 10 | op1st 8021 | . . . 4 ⊢ (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
12 | 4, 11 | eqtri 2763 | . . 3 ⊢ (1st ‘𝑊) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
13 | 12 | fveq2i 6910 | . 2 ⊢ (1st ‘(1st ‘𝑊)) = (1st ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) |
14 | hilablo 31189 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
15 | resexg 6047 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V |
17 | hvmulex 31040 | . . . 4 ⊢ ·ℎ ∈ V | |
18 | 17 | resex 6049 | . . 3 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ∈ V |
19 | 16, 18 | op1st 8021 | . 2 ⊢ (1st ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) = ( +ℎ ↾ (𝐻 × 𝐻)) |
20 | 2, 13, 19 | 3eqtrri 2768 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 1st c1st 8011 ℂcc 11151 ℝcr 11152 AbelOpcablo 30573 +𝑣 cpv 30614 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 normℎcno 30952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his3 31113 ax-his4 31114 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-grpo 30522 df-ablo 30574 df-va 30624 df-hnorm 30997 df-hvsub 31000 |
This theorem is referenced by: hhsst 31295 hhsssh2 31299 |
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