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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 1121 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝐶 ∈ 𝐾) |
3 | 2 | ralrimivva 3201 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 𝐶 ∈ 𝐾) |
4 | eqid 2733 | . . . 4 ⊢ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) | |
5 | 4 | fmpo 8051 | . . 3 ⊢ (∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 𝐶 ∈ 𝐾 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾) |
6 | 3, 5 | sylib 217 | . 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 21915 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
13 | 8, 9, 12 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
14 | 7, 13 | eqtr4id 2792 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝐾 ↑m (𝑁 × 𝑁))) |
15 | 14 | eleq2d 2820 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ (𝐾 ↑m (𝑁 × 𝑁)))) |
16 | 11 | fvexi 6903 | . . . 4 ⊢ 𝐾 ∈ V |
17 | 8, 8 | xpexd 7735 | . . . 4 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ V) |
18 | elmapg 8830 | . . . 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 588 | . . 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 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 × cxp 5674 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 ↑m cmap 8817 Fincfn 8936 Basecbs 17141 Mat cmat 21899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 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 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 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 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 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-hom 17218 df-cco 17219 df-0g 17384 df-prds 17390 df-pws 17392 df-sra 20778 df-rgmod 20779 df-dsmm 21279 df-frlm 21294 df-mat 21900 |
This theorem is referenced by: mpomatmul 21940 dmatmulcl 21994 scmatscmiddistr 22002 marrepcl 22058 marepvcl 22063 submabas 22072 mdetrsca2 22098 mdetr0 22099 mdetrlin2 22101 mdetralt2 22103 mdetero 22104 mdetunilem2 22107 mdetunilem5 22110 mdetunilem6 22111 maduf 22135 madutpos 22136 marep01ma 22154 mat2pmatbas 22220 mat2pmatghm 22224 cpm2mf 22246 m2cpminvid 22247 m2cpminvid2 22249 m2cpmfo 22250 decpmatcl 22261 decpmatmul 22266 pmatcollpw1 22270 pmatcollpw2 22272 monmatcollpw 22273 pmatcollpwlem 22274 pmatcollpw 22275 pmatcollpw3lem 22277 pmatcollpwscmatlem2 22284 pm2mpf1 22293 mply1topmatcl 22299 mp2pm2mplem2 22301 mp2pm2mplem4 22303 pm2mpghm 22310 lmatcl 32785 mdetpmtr1 32792 mdetpmtr2 32793 mdetpmtr12 32794 madjusmdetlem1 32796 madjusmdetlem3 32798 |
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