| Colors of
variables: wff
setvar class |
| Syntax hints:
→ wi 4 ∧ wa 396
= wceq 1541 ∈
wcel 2106 ‘cfv 6543
(class class class)co 7411 Fincfn 8941
Ringcrg 20127 Poly1cpl1 21920 Mat
cmat 22127 |
| This theorem was proved from axioms:
ax-mp 5 ax-1 6 ax-2 7
ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913
ax-6 1971 ax-7 2011 ax-8 2108
ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 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 397
df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 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-int 4951 df-iun 4999 df-iin 5000 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-se 5632 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 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-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 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-fzo 13632 df-seq 13971 df-hash 14295 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-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-sra 20930 df-rgmod 20931 df-dsmm 21506 df-frlm 21521 df-psr 21681 df-mpl 21683 df-opsr 21685 df-psr1 21923 df-ply1 21925 df-mamu 22106 df-mat 22128 |
| This theorem is referenced by: 1pmatscmul
22424 1elcpmat
22437 cpmatacl
22438 cpmatinvcl
22439 cpmatmcl
22441 cpmatsubgpmat
22442 cpmatsrgpmat
22443 mat2pmatghm
22452 mat2pmat1
22454 decpmatid
22492 decpmatmullem
22493 decpmatmul
22494 decpmatmulsumfsupp
22495 pmatcollpwfi
22504 pmatcollpw3fi1lem1
22508 idpm2idmp
22523 pm2mpghm
22538 pm2mpmhmlem1
22540 pm2mpmhmlem2
22541 pm2mpmhm
22542 pm2mprhm
22543 pm2mp
22547 chmatcl
22550 chmatval
22551 chpdmatlem0
22559 chpdmatlem1
22560 chpdmatlem2
22561 chpdmatlem3
22562 chmaidscmat
22570 chfacfisf
22576 chfacfscmulgsum
22582 chfacfpmmulcl
22583 chfacfpmmul0
22584 chfacfpmmulgsum
22586 chfacfpmmulgsum2
22587 cayhamlem1
22588 cpmadugsumlemF
22598 cpmadugsumfi
22599 cpmidgsum2
22601 cayhamlem4
22610 |
