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Mirrors > Home > MPE Home > Th. List > matbas2 | Structured version Visualization version GIF version |
Description: The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.) |
Ref | Expression |
---|---|
matbas2.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matbas2.k | ⊢ 𝐾 = (Base‘𝑅) |
Ref | Expression |
---|---|
matbas2 | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpfi 9313 | . . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin) | |
2 | 1 | anidms 566 | . . . 4 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ Fin) |
3 | 2 | anim1ci 615 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑅 ∈ 𝑉 ∧ (𝑁 × 𝑁) ∈ Fin)) |
4 | eqid 2724 | . . . 4 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) | |
5 | matbas2.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
6 | 4, 5 | frlmfibas 21625 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 × 𝑁) ∈ Fin) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
7 | 3, 6 | syl 17 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))) |
8 | matbas2.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
9 | 8, 4 | matbas 22235 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴)) |
10 | 7, 9 | eqtrd 2764 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 × cxp 5664 ‘cfv 6533 (class class class)co 7401 ↑m cmap 8816 Fincfn 8935 Basecbs 17143 freeLMod cfrlm 21609 Mat cmat 22229 |
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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-prds 17392 df-pws 17394 df-sra 21011 df-rgmod 21012 df-dsmm 21595 df-frlm 21610 df-mat 22230 |
This theorem is referenced by: matbas2i 22246 matbas2d 22247 matecl 22249 matvscacell 22260 matring 22267 matassa 22268 mpomatmul 22270 mat1 22271 mattposcl 22277 mat0dimbas0 22290 mat1dimelbas 22295 mat1f1o 22302 mavmulval 22369 mavmulcl 22371 mavmulass 22373 mavmumamul1 22379 mdetunilem9 22444 cramerimplem2 22508 mat2pmatmul 22555 decpmatmullem 22595 smatcl 33271 1smat1 33273 |
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