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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 5175 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4561 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | fvex 6658 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V | |
4 | map0e 8429 | . . . . . 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 8099 | . . . . 5 ⊢ 1o = {∅} | |
7 | 5, 6 | eqtrdi 2849 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅}) |
8 | 2, 7 | eleqtrrid 2897 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)) |
9 | 0elpw 5221 | . . . 4 ⊢ ∅ ∈ 𝒫 (Base‘𝑀) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ 𝒫 (Base‘𝑀)) |
11 | lincval 44818 | . . 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 6462 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅) |
15 | 14 | oveq2d 7151 | . . 3 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg ∅)) |
16 | eqid 2798 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
17 | 16 | gsum0 17886 | . . 3 ⊢ (𝑀 Σg ∅) = (0g‘𝑀) |
18 | 15, 17 | eqtrdi 2849 | . 2 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (0g‘𝑀)) |
19 | 12, 18 | eqtrd 2833 | 1 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 Σg cgsu 16706 linC clinc 44813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-map 8391 df-seq 13365 df-gsum 16708 df-linc 44815 |
This theorem is referenced by: lco0 44836 |
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