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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 2736 | . . 3 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
2 | 1 | vafval 29431 | . 2 ⊢ ( +𝑣 ‘𝑊) = (1st ‘(1st ‘𝑊)) |
3 | hhss.1 | . . . . 5 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
4 | 3 | fveq2i 6842 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
5 | opex 5419 | . . . . 5 ⊢ 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 ∈ V | |
6 | normf 29951 | . . . . . . 7 ⊢ normℎ: ℋ⟶ℝ | |
7 | ax-hilex 29827 | . . . . . . 7 ⊢ ℋ ∈ V | |
8 | fex 7172 | . . . . . . 7 ⊢ ((normℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → normℎ ∈ V) | |
9 | 6, 7, 8 | mp2an 690 | . . . . . 6 ⊢ normℎ ∈ V |
10 | 9 | resex 5983 | . . . . 5 ⊢ (normℎ ↾ 𝐻) ∈ V |
11 | 5, 10 | op1st 7925 | . . . 4 ⊢ (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
12 | 4, 11 | eqtri 2764 | . . 3 ⊢ (1st ‘𝑊) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
13 | 12 | fveq2i 6842 | . 2 ⊢ (1st ‘(1st ‘𝑊)) = (1st ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) |
14 | hilablo 29988 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
15 | resexg 5981 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V |
17 | hvmulex 29839 | . . . 4 ⊢ ·ℎ ∈ V | |
18 | 17 | resex 5983 | . . 3 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ∈ V |
19 | 16, 18 | op1st 7925 | . 2 ⊢ (1st ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) = ( +ℎ ↾ (𝐻 × 𝐻)) |
20 | 2, 13, 19 | 3eqtrri 2769 | 1 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3443 〈cop 4590 × cxp 5629 ↾ cres 5633 ⟶wf 6489 ‘cfv 6493 1st c1st 7915 ℂcc 11045 ℝcr 11046 AbelOpcablo 29372 +𝑣 cpv 29413 ℋchba 29747 +ℎ cva 29748 ·ℎ csm 29749 normℎcno 29751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-hilex 29827 ax-hfvadd 29828 ax-hvcom 29829 ax-hvass 29830 ax-hv0cl 29831 ax-hvaddid 29832 ax-hfvmul 29833 ax-hvmulid 29834 ax-hvdistr2 29837 ax-hvmul0 29838 ax-hfi 29907 ax-his1 29910 ax-his3 29912 ax-his4 29913 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-seq 13899 df-exp 13960 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-grpo 29321 df-ablo 29373 df-va 29423 df-hnorm 29796 df-hvsub 29799 |
This theorem is referenced by: hhsst 30094 hhsssh2 30098 |
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