![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mavmul0 | Structured version Visualization version GIF version |
Description: Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.) |
Ref | Expression |
---|---|
mavmul0.t | ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) |
Ref | Expression |
---|---|
mavmul0 | ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
2 | mavmul0.t | . . 3 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | eqid 2798 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2798 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | simpr 488 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
6 | 0fin 8730 | . . . . 5 ⊢ ∅ ∈ Fin | |
7 | eleq1 2877 | . . . . 5 ⊢ (𝑁 = ∅ → (𝑁 ∈ Fin ↔ ∅ ∈ Fin)) | |
8 | 6, 7 | mpbiri 261 | . . . 4 ⊢ (𝑁 = ∅ → 𝑁 ∈ Fin) |
9 | 8 | adantr 484 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin) |
10 | 0ex 5175 | . . . . 5 ⊢ ∅ ∈ V | |
11 | snidg 4559 | . . . . 5 ⊢ (∅ ∈ V → ∅ ∈ {∅}) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ {∅}) |
13 | oveq1 7142 | . . . . . . 7 ⊢ (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) | |
14 | 13 | adantr 484 | . . . . . 6 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) |
15 | 14 | fveq2d 6649 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅))) |
16 | mat0dimbas0 21071 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) | |
17 | 16 | adantl 485 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(∅ Mat 𝑅)) = {∅}) |
18 | 15, 17 | eqtrd 2833 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = {∅}) |
19 | 12, 18 | eleqtrrd 2893 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ (Base‘(𝑁 Mat 𝑅))) |
20 | eqidd 2799 | . . . . . 6 ⊢ (𝑁 = ∅ → ∅ = ∅) | |
21 | el1o 8107 | . . . . . 6 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
22 | 20, 21 | sylibr 237 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ ∈ 1o) |
23 | oveq2 7143 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = ((Base‘𝑅) ↑m ∅)) | |
24 | fvex 6658 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
25 | map0e 8429 | . . . . . . 7 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑m ∅) = 1o) | |
26 | 24, 25 | mp1i 13 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m ∅) = 1o) |
27 | 23, 26 | eqtrd 2833 | . . . . 5 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = 1o) |
28 | 22, 27 | eleqtrrd 2893 | . . . 4 ⊢ (𝑁 = ∅ → ∅ ∈ ((Base‘𝑅) ↑m 𝑁)) |
29 | 28 | adantr 484 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ ((Base‘𝑅) ↑m 𝑁)) |
30 | 1, 2, 3, 4, 5, 9, 19, 29 | mavmulval 21150 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) |
31 | mpteq1 5118 | . . . 4 ⊢ (𝑁 = ∅ → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) | |
32 | 31 | adantr 484 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) |
33 | mpt0 6462 | . . 3 ⊢ (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅ | |
34 | 32, 33 | eqtrdi 2849 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅) |
35 | 30, 34 | eqtrd 2833 | 1 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 〈cop 4531 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 ↑m cmap 8389 Fincfn 8492 Basecbs 16475 .rcmulr 16558 Σg cgsu 16706 Mat cmat 21012 maVecMul cmvmul 21145 |
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 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-nel 3092 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-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-ord 6162 df-on 6163 df-lim 6164 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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-pws 16715 df-sra 19937 df-rgmod 19938 df-dsmm 20421 df-frlm 20436 df-mat 21013 df-mvmul 21146 |
This theorem is referenced by: mavmul0g 21158 cramer0 21295 |
Copyright terms: Public domain | W3C validator |