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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 4790 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4208 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | fvex 6201 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V | |
4 | map0e 7895 | . . . . . 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 7572 | . . . . 5 ⊢ 1𝑜 = {∅} | |
7 | 5, 6 | syl6eq 2672 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = {∅}) |
8 | 2, 7 | syl5eleqr 2708 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅)) |
9 | 0elpw 4834 | . . . 4 ⊢ ∅ ∈ 𝒫 (Base‘𝑀) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ 𝒫 (Base‘𝑀)) |
11 | lincval 42198 | . . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) ∧ ∅ ∈ 𝒫 (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) | |
12 | 8, 10, 11 | mpd3an23 1426 | . 2 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
13 | mpt0 6021 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅) |
15 | 14 | oveq2d 6666 | . . 3 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg ∅)) |
16 | eqid 2622 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
17 | 16 | gsum0 17278 | . . 3 ⊢ (𝑀 Σg ∅) = (0g‘𝑀) |
18 | 15, 17 | syl6eq 2672 | . 2 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (0g‘𝑀)) |
19 | 12, 18 | eqtrd 2656 | 1 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 ↑𝑚 cmap 7857 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 Σg cgsu 16101 linC clinc 42193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-map 7859 df-seq 12802 df-gsum 16103 df-linc 42195 |
This theorem is referenced by: lco0 42216 |
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