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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 2778 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
2 | mavmul0.t | . . 3 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
3 | eqid 2778 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2778 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | simpr 479 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
6 | 0fin 8476 | . . . . 5 ⊢ ∅ ∈ Fin | |
7 | eleq1 2847 | . . . . 5 ⊢ (𝑁 = ∅ → (𝑁 ∈ Fin ↔ ∅ ∈ Fin)) | |
8 | 6, 7 | mpbiri 250 | . . . 4 ⊢ (𝑁 = ∅ → 𝑁 ∈ Fin) |
9 | 8 | adantr 474 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin) |
10 | 0ex 5026 | . . . . 5 ⊢ ∅ ∈ V | |
11 | snidg 4428 | . . . . 5 ⊢ (∅ ∈ V → ∅ ∈ {∅}) | |
12 | 10, 11 | mp1i 13 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ {∅}) |
13 | oveq1 6929 | . . . . . . 7 ⊢ (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) | |
14 | 13 | adantr 474 | . . . . . 6 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) |
15 | 14 | fveq2d 6450 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅))) |
16 | mat0dimbas0 20677 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) | |
17 | 16 | adantl 475 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(∅ Mat 𝑅)) = {∅}) |
18 | 15, 17 | eqtrd 2814 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = {∅}) |
19 | 12, 18 | eleqtrrd 2862 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ (Base‘(𝑁 Mat 𝑅))) |
20 | eqidd 2779 | . . . . . 6 ⊢ (𝑁 = ∅ → ∅ = ∅) | |
21 | el1o 7863 | . . . . . 6 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
22 | 20, 21 | sylibr 226 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ ∈ 1o) |
23 | oveq2 6930 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑𝑚 𝑁) = ((Base‘𝑅) ↑𝑚 ∅)) | |
24 | fvex 6459 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
25 | map0e 8179 | . . . . . . 7 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑𝑚 ∅) = 1o) | |
26 | 24, 25 | mp1i 13 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑𝑚 ∅) = 1o) |
27 | 23, 26 | eqtrd 2814 | . . . . 5 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑𝑚 𝑁) = 1o) |
28 | 22, 27 | eleqtrrd 2862 | . . . 4 ⊢ (𝑁 = ∅ → ∅ ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
29 | 28 | adantr 474 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ ((Base‘𝑅) ↑𝑚 𝑁)) |
30 | 1, 2, 3, 4, 5, 9, 19, 29 | mavmulval 20756 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) |
31 | mpteq1 4972 | . . . 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 474 | . . 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 6267 | . . 3 ⊢ (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅ | |
34 | 32, 33 | syl6eq 2830 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅) |
35 | 30, 34 | eqtrd 2814 | 1 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 {csn 4398 〈cop 4404 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 1oc1o 7836 ↑𝑚 cmap 8140 Fincfn 8241 Basecbs 16255 .rcmulr 16339 Σg cgsu 16487 Mat cmat 20617 maVecMul cmvmul 20751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-prds 16494 df-pws 16496 df-sra 19569 df-rgmod 19570 df-dsmm 20475 df-frlm 20490 df-mat 20618 df-mvmul 20752 |
This theorem is referenced by: mavmul0g 20764 cramer0 20903 |
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