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Mirrors > Home > MPE Home > Th. List > matbas2 | Structured version Unicode 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 |
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matbas2.k |
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Ref | Expression |
---|---|
matbas2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpfi 7697 |
. . . . . 6
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2 | 1 | anidms 645 |
. . . . 5
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3 | 2 | anim2i 569 |
. . . 4
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4 | 3 | ancoms 453 |
. . 3
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5 | eqid 2454 |
. . . 4
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6 | matbas2.k |
. . . 4
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7 | 5, 6 | frlmfibas 18317 |
. . 3
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8 | 4, 7 | syl 16 |
. 2
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9 | matbas2.a |
. . 3
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10 | 9, 5 | matbas 18442 |
. 2
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11 | 8, 10 | eqtrd 2495 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-cnex 9452 ax-resscn 9453 ax-1cn 9454 ax-icn 9455 ax-addcl 9456 ax-addrcl 9457 ax-mulcl 9458 ax-mulrcl 9459 ax-mulcom 9460 ax-addass 9461 ax-mulass 9462 ax-distr 9463 ax-i2m1 9464 ax-1ne0 9465 ax-1rid 9466 ax-rnegex 9467 ax-rrecex 9468 ax-cnre 9469 ax-pre-lttri 9470 ax-pre-lttrn 9471 ax-pre-ltadd 9472 ax-pre-mulgt0 9473 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-ot 3997 df-uni 4203 df-int 4240 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-1st 6690 df-2nd 6691 df-supp 6804 df-recs 6945 df-rdg 6979 df-1o 7033 df-oadd 7037 df-er 7214 df-map 7329 df-ixp 7377 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-fsupp 7735 df-sup 7805 df-pnf 9534 df-mnf 9535 df-xr 9536 df-ltxr 9537 df-le 9538 df-sub 9711 df-neg 9712 df-nn 10437 df-2 10494 df-3 10495 df-4 10496 df-5 10497 df-6 10498 df-7 10499 df-8 10500 df-9 10501 df-10 10502 df-n0 10694 df-z 10761 df-dec 10870 df-uz 10976 df-fz 11558 df-struct 14297 df-ndx 14298 df-slot 14299 df-base 14300 df-sets 14301 df-ress 14302 df-plusg 14373 df-mulr 14374 df-sca 14376 df-vsca 14377 df-ip 14378 df-tset 14379 df-ple 14380 df-ds 14382 df-hom 14384 df-cco 14385 df-0g 14502 df-prds 14508 df-pws 14510 df-sra 17379 df-rgmod 17380 df-dsmm 18285 df-frlm 18300 df-mat 18410 |
This theorem is referenced by: matbas2i 18451 matbas2d 18452 mat0dimbas0 18453 matecl 18454 matrng 18459 matassa 18460 mat1 18464 mattposcl 18467 mavmulval 18486 mavmulcl 18488 mavmulass 18490 mavmumamul1 18496 mdetunilem9 18561 cramerimplem2 18625 matvscacell 31041 mpt2matmul 31048 mat1dimelbas 31053 mat2pmatmul 31240 decpmatmullem 31278 |
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