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Mirrors > Home > MPE Home > Th. List > smfval | Structured version Visualization version GIF version |
Description: Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
smfval.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
smfval | ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfval.4 | . 2 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
2 | df-sm 26836 | . . . . 5 ⊢ ·𝑠OLD = (2nd ∘ 1st ) | |
3 | 2 | fveq1i 6104 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) = ((2nd ∘ 1st )‘𝑈) |
4 | fo1st 7079 | . . . . . 6 ⊢ 1st :V–onto→V | |
5 | fof 6028 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
7 | fvco3 6185 | . . . . 5 ⊢ ((1st :V⟶V ∧ 𝑈 ∈ V) → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st ‘𝑈))) | |
8 | 6, 7 | mpan 702 | . . . 4 ⊢ (𝑈 ∈ V → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st ‘𝑈))) |
9 | 3, 8 | syl5eq 2656 | . . 3 ⊢ (𝑈 ∈ V → ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈))) |
10 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( ·𝑠OLD ‘𝑈) = ∅) | |
11 | fvprc 6097 | . . . . . 6 ⊢ (¬ 𝑈 ∈ V → (1st ‘𝑈) = ∅) | |
12 | 11 | fveq2d 6107 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → (2nd ‘(1st ‘𝑈)) = (2nd ‘∅)) |
13 | 2nd0 7066 | . . . . 5 ⊢ (2nd ‘∅) = ∅ | |
14 | 12, 13 | syl6req 2661 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ∅ = (2nd ‘(1st ‘𝑈))) |
15 | 10, 14 | eqtrd 2644 | . . 3 ⊢ (¬ 𝑈 ∈ V → ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈))) |
16 | 9, 15 | pm2.61i 175 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = (2nd ‘(1st ‘𝑈)) |
17 | 1, 16 | eqtri 2632 | 1 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ∘ ccom 5042 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 ·𝑠OLD cns 26826 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-sm 26836 |
This theorem is referenced by: nvvop 26848 nvsf 26858 nvscl 26865 nvsid 26866 nvsass 26867 nvdi 26869 nvdir 26870 nv2 26871 nv0 26876 nvsz 26877 nvinv 26878 nvtri 26909 cnnvs 26919 phop 27057 phpar 27063 ipdirilem 27068 h2hsm 27216 hhsssm 27499 |
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