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Mirrors > Home > HSE Home > Th. List > hhsssm | Structured version Visualization version GIF version |
Description: The scalar multiplication 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 |
---|---|
hhsssm | ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
2 | 1 | smfval 26844 | . 2 ⊢ ( ·𝑠OLD ‘𝑊) = (2nd ‘(1st ‘𝑊)) |
3 | hhss.1 | . . . . 5 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
4 | 3 | fveq2i 6106 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) |
5 | opex 4859 | . . . . 5 ⊢ 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 ∈ V | |
6 | normf 27364 | . . . . . . 7 ⊢ normℎ: ℋ⟶ℝ | |
7 | ax-hilex 27240 | . . . . . . 7 ⊢ ℋ ∈ V | |
8 | fex 6394 | . . . . . . 7 ⊢ ((normℎ: ℋ⟶ℝ ∧ ℋ ∈ V) → normℎ ∈ V) | |
9 | 6, 7, 8 | mp2an 704 | . . . . . 6 ⊢ normℎ ∈ V |
10 | 9 | resex 5363 | . . . . 5 ⊢ (normℎ ↾ 𝐻) ∈ V |
11 | 5, 10 | op1st 7067 | . . . 4 ⊢ (1st ‘〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
12 | 4, 11 | eqtri 2632 | . . 3 ⊢ (1st ‘𝑊) = 〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉 |
13 | 12 | fveq2i 6106 | . 2 ⊢ (2nd ‘(1st ‘𝑊)) = (2nd ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) |
14 | hilablo 27401 | . . . 4 ⊢ +ℎ ∈ AbelOp | |
15 | resexg 5362 | . . . 4 ⊢ ( +ℎ ∈ AbelOp → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ V |
17 | hvmulex 27252 | . . . 4 ⊢ ·ℎ ∈ V | |
18 | 17 | resex 5363 | . . 3 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) ∈ V |
19 | 16, 18 | op2nd 7068 | . 2 ⊢ (2nd ‘〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉) = ( ·ℎ ↾ (ℂ × 𝐻)) |
20 | 2, 13, 19 | 3eqtrri 2637 | 1 ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 ℂcc 9813 ℝcr 9814 AbelOpcablo 26782 ·𝑠OLD cns 26826 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 normℎcno 27164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-grpo 26731 df-ablo 26783 df-sm 26836 df-hnorm 27209 df-hvsub 27212 |
This theorem is referenced by: hhsst 27507 hhsssh2 27511 |
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