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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 2737 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
| 2 | mavmul0.t | . . 3 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
| 3 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 4 | eqid 2737 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 5 | simpr 484 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
| 6 | 0fi 9082 | . . . . 5 ⊢ ∅ ∈ Fin | |
| 7 | eleq1 2829 | . . . . 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 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 11 | snidg 4660 | . . . . 5 ⊢ (∅ ∈ V → ∅ ∈ {∅}) | |
| 12 | 10, 11 | mp1i 13 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ {∅}) | 
| 13 | oveq1 7438 | . . . . . . 7 ⊢ (𝑁 = ∅ → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) | |
| 14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑁 Mat 𝑅) = (∅ Mat 𝑅)) | 
| 15 | 14 | fveq2d 6910 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = (Base‘(∅ Mat 𝑅))) | 
| 16 | mat0dimbas0 22472 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (Base‘(∅ Mat 𝑅)) = {∅}) | |
| 17 | 16 | adantl 481 | . . . . 5 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(∅ Mat 𝑅)) = {∅}) | 
| 18 | 15, 17 | eqtrd 2777 | . . . 4 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 Mat 𝑅)) = {∅}) | 
| 19 | 12, 18 | eleqtrrd 2844 | . . 3 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → ∅ ∈ (Base‘(𝑁 Mat 𝑅))) | 
| 20 | eqidd 2738 | . . . . . 6 ⊢ (𝑁 = ∅ → ∅ = ∅) | |
| 21 | el1o 8533 | . . . . . 6 ⊢ (∅ ∈ 1o ↔ ∅ = ∅) | |
| 22 | 20, 21 | sylibr 234 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ ∈ 1o) | 
| 23 | oveq2 7439 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = ((Base‘𝑅) ↑m ∅)) | |
| 24 | fvex 6919 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
| 25 | map0e 8922 | . . . . . . 7 ⊢ ((Base‘𝑅) ∈ V → ((Base‘𝑅) ↑m ∅) = 1o) | |
| 26 | 24, 25 | mp1i 13 | . . . . . 6 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m ∅) = 1o) | 
| 27 | 23, 26 | eqtrd 2777 | . . . . 5 ⊢ (𝑁 = ∅ → ((Base‘𝑅) ↑m 𝑁) = 1o) | 
| 28 | 22, 27 | eleqtrrd 2844 | . . . 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 22551 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗)))))) | 
| 31 | mpteq1 5235 | . . . 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 6710 | . . 3 ⊢ (𝑖 ∈ ∅ ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅ | |
| 34 | 32, 33 | eqtrdi 2793 | . 2 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (𝑖 ∈ 𝑁 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖∅𝑗)(.r‘𝑅)(∅‘𝑗))))) = ∅) | 
| 35 | 30, 34 | eqtrd 2777 | 1 ⊢ ((𝑁 = ∅ ∧ 𝑅 ∈ 𝑉) → (∅ · ∅) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 ↑m cmap 8866 Fincfn 8985 Basecbs 17247 .rcmulr 17298 Σg cgsu 17485 Mat cmat 22411 maVecMul cmvmul 22546 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-prds 17492 df-pws 17494 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-mat 22412 df-mvmul 22547 | 
| This theorem is referenced by: mavmul0g 22559 cramer0 22696 | 
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