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Mirrors > Home > MPE Home > Th. List > nvsf | Structured version Visualization version GIF version |
Description: Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvsf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvsf.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvsf | ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 29846 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2733 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 29834 | . . 3 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvsf.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 29836 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvsf.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 29835 | . . 3 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
9 | 4, 6, 8 | vcsm 29793 | . 2 ⊢ ((1st ‘𝑈) ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋) |
10 | 2, 9 | syl 17 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 × cxp 5673 ⟶wf 6536 ‘cfv 6540 1st c1st 7968 ℂcc 11104 CVecOLDcvc 29789 NrmCVeccnv 29815 +𝑣 cpv 29816 BaseSetcba 29817 ·𝑠OLD cns 29818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-1st 7970 df-2nd 7971 df-vc 29790 df-nv 29823 df-va 29826 df-ba 29827 df-sm 29828 df-0v 29829 df-nmcv 29831 |
This theorem is referenced by: nvinvfval 29871 smcnlem 29928 ssps 29961 hlmulf 30135 |
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