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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 1118 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝐶 ∈ 𝐾) |
3 | 2 | ralrimivva 3198 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 𝐶 ∈ 𝐾) |
4 | eqid 2730 | . . . 4 ⊢ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) | |
5 | 4 | fmpo 8056 | . . 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 22143 | . . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
13 | 8, 9, 12 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝐾 ↑m (𝑁 × 𝑁)) = (Base‘𝐴)) |
14 | 7, 13 | eqtr4id 2789 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝐾 ↑m (𝑁 × 𝑁))) |
15 | 14 | eleq2d 2817 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ (𝐾 ↑m (𝑁 × 𝑁)))) |
16 | 11 | fvexi 6904 | . . . 4 ⊢ 𝐾 ∈ V |
17 | 8, 8 | xpexd 7740 | . . . 4 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ V) |
18 | elmapg 8835 | . . . 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 585 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ (𝐾 ↑m (𝑁 × 𝑁)) ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾)) |
20 | 15, 19 | bitrd 278 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵 ↔ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶):(𝑁 × 𝑁)⟶𝐾)) |
21 | 6, 20 | mpbird 256 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ 𝐶) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 Vcvv 3472 × cxp 5673 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 ↑m cmap 8822 Fincfn 8941 Basecbs 17148 Mat cmat 22127 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-prds 17397 df-pws 17399 df-sra 20930 df-rgmod 20931 df-dsmm 21506 df-frlm 21521 df-mat 22128 |
This theorem is referenced by: mpomatmul 22168 dmatmulcl 22222 scmatscmiddistr 22230 marrepcl 22286 marepvcl 22291 submabas 22300 mdetrsca2 22326 mdetr0 22327 mdetrlin2 22329 mdetralt2 22331 mdetero 22332 mdetunilem2 22335 mdetunilem5 22338 mdetunilem6 22339 maduf 22363 madutpos 22364 marep01ma 22382 mat2pmatbas 22448 mat2pmatghm 22452 cpm2mf 22474 m2cpminvid 22475 m2cpminvid2 22477 m2cpmfo 22478 decpmatcl 22489 decpmatmul 22494 pmatcollpw1 22498 pmatcollpw2 22500 monmatcollpw 22501 pmatcollpwlem 22502 pmatcollpw 22503 pmatcollpw3lem 22505 pmatcollpwscmatlem2 22512 pm2mpf1 22521 mply1topmatcl 22527 mp2pm2mplem2 22529 mp2pm2mplem4 22531 pm2mpghm 22538 lmatcl 33094 mdetpmtr1 33101 mdetpmtr2 33102 mdetpmtr12 33103 madjusmdetlem1 33105 madjusmdetlem3 33107 |
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