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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 8054 | . . 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 21923 | . . . . . 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 6906 | . . . 4 ⊢ 𝐾 ∈ V |
17 | 8, 8 | xpexd 7738 | . . . 4 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ V) |
18 | elmapg 8833 | . . . 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 5675 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 ↑m cmap 8820 Fincfn 8939 Basecbs 17144 Mat cmat 21907 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-prds 17393 df-pws 17395 df-sra 20785 df-rgmod 20786 df-dsmm 21287 df-frlm 21302 df-mat 21908 |
This theorem is referenced by: mpomatmul 21948 dmatmulcl 22002 scmatscmiddistr 22010 marrepcl 22066 marepvcl 22071 submabas 22080 mdetrsca2 22106 mdetr0 22107 mdetrlin2 22109 mdetralt2 22111 mdetero 22112 mdetunilem2 22115 mdetunilem5 22118 mdetunilem6 22119 maduf 22143 madutpos 22144 marep01ma 22162 mat2pmatbas 22228 mat2pmatghm 22232 cpm2mf 22254 m2cpminvid 22255 m2cpminvid2 22257 m2cpmfo 22258 decpmatcl 22269 decpmatmul 22274 pmatcollpw1 22278 pmatcollpw2 22280 monmatcollpw 22281 pmatcollpwlem 22282 pmatcollpw 22283 pmatcollpw3lem 22285 pmatcollpwscmatlem2 22292 pm2mpf1 22301 mply1topmatcl 22307 mp2pm2mplem2 22309 mp2pm2mplem4 22311 pm2mpghm 22318 lmatcl 32796 mdetpmtr1 32803 mdetpmtr2 32804 mdetpmtr12 32805 madjusmdetlem1 32807 madjusmdetlem3 32809 |
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