Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 5213 | . . . . 5 ⊢ ∅ ∈ V | |
2 | 1 | snid 4603 | . . . 4 ⊢ ∅ ∈ {∅} |
3 | fvex 6685 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V | |
4 | map0e 8448 | . . . . . 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 8118 | . . . . 5 ⊢ 1o = {∅} | |
7 | 5, 6 | syl6eq 2874 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅}) |
8 | 2, 7 | eleqtrrid 2922 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)) |
9 | 0elpw 5258 | . . . 4 ⊢ ∅ ∈ 𝒫 (Base‘𝑀) | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑀 ∈ 𝑋 → ∅ ∈ 𝒫 (Base‘𝑀)) |
11 | lincval 44471 | . . 3 ⊢ ((𝑀 ∈ 𝑋 ∧ ∅ ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅) ∧ ∅ ∈ 𝒫 (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) | |
12 | 8, 10, 11 | mpd3an23 1459 | . 2 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) |
13 | mpt0 6492 | . . . . 5 ⊢ (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑀 ∈ 𝑋 → (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = ∅) |
15 | 14 | oveq2d 7174 | . . 3 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg ∅)) |
16 | eqid 2823 | . . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
17 | 16 | gsum0 17896 | . . 3 ⊢ (𝑀 Σg ∅) = (0g‘𝑀) |
18 | 15, 17 | syl6eq 2874 | . 2 ⊢ (𝑀 ∈ 𝑋 → (𝑀 Σg (𝑣 ∈ ∅ ↦ ((∅‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (0g‘𝑀)) |
19 | 12, 18 | eqtrd 2858 | 1 ⊢ (𝑀 ∈ 𝑋 → (∅( linC ‘𝑀)∅) = (0g‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 𝒫 cpw 4541 {csn 4569 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 ↑m cmap 8408 Basecbs 16485 Scalarcsca 16570 ·𝑠 cvsca 16571 0gc0g 16715 Σg cgsu 16716 linC clinc 44466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-map 8410 df-seq 13373 df-gsum 16718 df-linc 44468 |
This theorem is referenced by: lco0 44489 |
Copyright terms: Public domain | W3C validator |