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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 2724 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
2 | mavmul0.t | . . 3 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | eqid 2724 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2724 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | simpr 484 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
6 | 0fin 9168 | . . . . 5 ⊢ ∅ ∈ Fin | |
7 | eleq1 2813 | . . . . 5 ⊢ (𝑁 = ∅ → (𝑁 ∈ Fin ↔ ∅ ∈ Fin)) | |
8 | 6, 7 | mpbiri 258 | . . . 4 ⊢ (𝑁 = ∅ → 𝑁 ∈ Fin) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin) |
10 | 0ex 5298 | . . . . 5 ⊢ ∅ ∈ V | |
11 | snidg 4655 | . . . . 5 ⊢ (∅ ∈ V → ∅ ∈ {∅}) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ {∅}) |
13 | oveq1 7409 | . . . . . . 7 ⊢ (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) | |
14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) |
15 | 14 | fveq2d 6886 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅))) |
16 | mat0dimbas0 22292 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) | |
17 | 16 | adantl 481 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(∅ Mat 𝑅)) = {∅}) |
18 | 15, 17 | eqtrd 2764 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = {∅}) |
19 | 12, 18 | eleqtrrd 2828 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ (Base‘(𝑁 Mat 𝑅))) |
20 | eqidd 2725 | . . . . . 6 ⊢ (𝑁 = ∅ → ∅ = ∅) | |
21 | el1o 8491 | . . . . . 6 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
22 | 20, 21 | sylibr 233 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ ∈ 1o) |
23 | oveq2 7410 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = ((Base‘𝑅) ↑m ∅)) | |
24 | fvex 6895 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
25 | map0e 8873 | . . . . . . 7 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑m ∅) = 1o) | |
26 | 24, 25 | mp1i 13 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m ∅) = 1o) |
27 | 23, 26 | eqtrd 2764 | . . . . 5 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = 1o) |
28 | 22, 27 | eleqtrrd 2828 | . . . 4 ⊢ (𝑁 = ∅ → ∅ ∈ ((Base‘𝑅) ↑m 𝑁)) |
29 | 28 | adantr 480 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ ((Base‘𝑅) ↑m 𝑁)) |
30 | 1, 2, 3, 4, 5, 9, 19, 29 | mavmulval 22371 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) |
31 | mpteq1 5232 | . . . 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 480 | . . 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 6683 | . . 3 ⊢ (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅ | |
34 | 32, 33 | eqtrdi 2780 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅) |
35 | 30, 34 | eqtrd 2764 | 1 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4315 {csn 4621 〈cop 4627 ↦ cmpt 5222 ‘cfv 6534 (class class class)co 7402 1oc1o 8455 ↑m cmap 8817 Fincfn 8936 Basecbs 17145 .rcmulr 17199 Σg cgsu 17387 Mat cmat 22231 maVecMul cmvmul 22366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-ot 4630 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-hom 17222 df-cco 17223 df-0g 17388 df-prds 17394 df-pws 17396 df-sra 21013 df-rgmod 21014 df-dsmm 21597 df-frlm 21612 df-mat 22232 df-mvmul 22367 |
This theorem is referenced by: mavmul0g 22379 cramer0 22516 |
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