|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > matbas2d | Structured version Visualization version GIF version | ||
| Description: The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.) | 
| Ref | Expression | 
|---|---|
| matbas2.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) | 
| matbas2.k | ⊢ 𝐾 = (Base‘𝑅) | 
| matbas2i.b | ⊢ 𝐵 = (Base‘𝐴) | 
| matbas2d.n | ⊢ (𝜑 → 𝑁 ∈ Fin) | 
| matbas2d.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) | 
| matbas2d.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝐶 ∈ 𝐾) | 
| Ref | Expression | 
|---|---|
| matbas2d | ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | matbas2d.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁) → 𝐶 ∈ 𝐾) | |
| 2 | 1 | 3expb 1120 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝐶 ∈ 𝐾) | 
| 3 | 2 | ralrimivva 3201 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 𝐶 ∈ 𝐾) | 
| 4 | eqid 2736 | . . . 4 ⊢ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) | |
| 5 | 4 | fmpo 8094 | . . 3 ⊢ (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 𝐶 ∈ 𝐾 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾) | 
| 6 | 3, 5 | sylib 218 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾) | 
| 7 | matbas2i.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
| 8 | matbas2d.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
| 9 | matbas2d.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
| 10 | matbas2.a | . . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 11 | matbas2.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑅) | |
| 12 | 10, 11 | matbas2 22428 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) | 
| 13 | 8, 9, 12 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) | 
| 14 | 7, 13 | eqtr4id 2795 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝐾 ↑m (𝑁 × 𝑁))) | 
| 15 | 14 | eleq2d 2826 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ (𝐾 ↑m (𝑁 × 𝑁)))) | 
| 16 | 11 | fvexi 6919 | . . . 4 ⊢ 𝐾 ∈ V | 
| 17 | 8, 8 | xpexd 7772 | . . . 4 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ V) | 
| 18 | elmapg 8880 | . . . 4 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ V) → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾)) | |
| 19 | 16, 17, 18 | sylancr 587 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾)) | 
| 20 | 15, 19 | bitrd 279 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾)) | 
| 21 | 6, 20 | mpbird 257 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 × cxp 5682 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ↑m cmap 8867 Fincfn 8986 Basecbs 17248 Mat cmat 22412 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-prds 17493 df-pws 17495 df-sra 21173 df-rgmod 21174 df-dsmm 21753 df-frlm 21768 df-mat 22413 | 
| This theorem is referenced by: mpomatmul 22453 dmatmulcl 22507 scmatscmiddistr 22515 marrepcl 22571 marepvcl 22576 submabas 22585 mdetrsca2 22611 mdetr0 22612 mdetrlin2 22614 mdetralt2 22616 mdetero 22617 mdetunilem2 22620 mdetunilem5 22623 mdetunilem6 22624 maduf 22648 madutpos 22649 marep01ma 22667 mat2pmatbas 22733 mat2pmatghm 22737 cpm2mf 22759 m2cpminvid 22760 m2cpminvid2 22762 m2cpmfo 22763 decpmatcl 22774 decpmatmul 22779 pmatcollpw1 22783 pmatcollpw2 22785 monmatcollpw 22786 pmatcollpwlem 22787 pmatcollpw 22788 pmatcollpw3lem 22790 pmatcollpwscmatlem2 22797 pm2mpf1 22806 mply1topmatcl 22812 mp2pm2mplem2 22814 mp2pm2mplem4 22816 pm2mpghm 22823 lmatcl 33816 mdetpmtr1 33823 mdetpmtr2 33824 mdetpmtr12 33825 madjusmdetlem1 33827 madjusmdetlem3 33829 | 
| Copyright terms: Public domain | W3C validator |