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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 5325 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4684 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | fvex 6935 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V | |
4 | map0e 8942 | . . . . . 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 8531 | . . . . 5 ⊢ 1o = {∅} | |
7 | 5, 6 | eqtrdi 2796 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅}) |
8 | 2, 7 | eleqtrrid 2851 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)) |
9 | 0elpw 5374 | . . . 4 ⊢ ∅ ∈ 𝒫 (Base‘𝑀) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ 𝒫 (Base‘𝑀)) |
11 | lincval 48140 | . . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅) ∧ ∅ ∈ 𝒫 (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) | |
12 | 8, 10, 11 | mpd3an23 1463 | . 2 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
13 | mpt0 6724 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅) |
15 | 14 | oveq2d 7466 | . . 3 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg ∅)) |
16 | eqid 2740 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
17 | 16 | gsum0 18724 | . . 3 ⊢ (𝑀 Σg ∅) = (0g‘𝑀) |
18 | 15, 17 | eqtrdi 2796 | . 2 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (0g‘𝑀)) |
19 | 12, 18 | eqtrd 2780 | 1 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ∅c0 4352 𝒫 cpw 4622 {csn 4648 ↦ cmpt 5249 ‘cfv 6575 (class class class)co 7450 1oc1o 8517 ↑m cmap 8886 Basecbs 17260 Scalarcsca 17316 ·𝑠 cvsca 17317 0gc0g 17501 Σg cgsu 17502 linC clinc 48135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-map 8888 df-seq 14055 df-gsum 17504 df-linc 48137 |
This theorem is referenced by: lco0 48158 |
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