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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincval0 | Structured version Visualization version GIF version |
Description: The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.) |
Ref | Expression |
---|---|
lincval0 | ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5312 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4669 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | fvex 6914 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V | |
4 | map0e 8911 | . . . . . 6 ⊢ ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o) |
6 | df1o2 8503 | . . . . 5 ⊢ 1o = {∅} | |
7 | 5, 6 | eqtrdi 2782 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅}) |
8 | 2, 7 | eleqtrrid 2833 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)) |
9 | 0elpw 5360 | . . . 4 ⊢ ∅ ∈ 𝒫 (Base‘𝑀) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ 𝒫 (Base‘𝑀)) |
11 | lincval 47792 | . . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅) ∧ ∅ ∈ 𝒫 (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) | |
12 | 8, 10, 11 | mpd3an23 1460 | . 2 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
13 | mpt0 6703 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅) |
15 | 14 | oveq2d 7440 | . . 3 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg ∅)) |
16 | eqid 2726 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
17 | 16 | gsum0 18677 | . . 3 ⊢ (𝑀 Σg ∅) = (0g‘𝑀) |
18 | 15, 17 | eqtrdi 2782 | . 2 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (0g‘𝑀)) |
19 | 12, 18 | eqtrd 2766 | 1 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 𝒫 cpw 4607 {csn 4633 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 1oc1o 8489 ↑m cmap 8855 Basecbs 17213 Scalarcsca 17269 ·𝑠 cvsca 17270 0gc0g 17454 Σg cgsu 17455 linC clinc 47787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-map 8857 df-seq 14022 df-gsum 17457 df-linc 47789 |
This theorem is referenced by: lco0 47810 |
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