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Mirrors > Home > MPE Home > Th. List > matvscacell | Structured version Visualization version Unicode version |
Description: Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.) |
Ref | Expression |
---|---|
matplusgcell.a |
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matplusgcell.b |
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matvscacell.k |
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matvscacell.v |
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matvscacell.t |
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Ref | Expression |
---|---|
matvscacell |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | matplusgcell.a |
. . . . 5
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2 | matplusgcell.b |
. . . . 5
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3 | matvscacell.k |
. . . . 5
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4 | matvscacell.v |
. . . . 5
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5 | matvscacell.t |
. . . . 5
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6 | eqid 2451 |
. . . . 5
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7 | 1, 2, 3, 4, 5, 6 | matvsca2 19453 |
. . . 4
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8 | 7 | oveqd 6307 |
. . 3
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9 | 8 | 3ad2ant2 1030 |
. 2
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10 | df-ov 6293 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | a1i 11 |
. 2
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12 | opelxpi 4866 |
. . . 4
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13 | 12 | 3ad2ant3 1031 |
. . 3
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14 | 1, 2 | matrcl 19437 |
. . . . . . . 8
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15 | 14 | simpld 461 |
. . . . . . 7
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16 | 15 | adantl 468 |
. . . . . 6
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17 | 16 | 3ad2ant2 1030 |
. . . . 5
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18 | xpfi 7842 |
. . . . 5
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19 | 17, 17, 18 | syl2anc 667 |
. . . 4
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20 | simp2l 1034 |
. . . 4
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21 | 2 | eleq2i 2521 |
. . . . . . . . 9
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22 | 21 | biimpi 198 |
. . . . . . . 8
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23 | 22 | adantl 468 |
. . . . . . 7
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24 | 23 | 3ad2ant2 1030 |
. . . . . 6
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25 | simp1 1008 |
. . . . . . 7
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26 | eqid 2451 |
. . . . . . . 8
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27 | 1, 26 | matbas2 19446 |
. . . . . . 7
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28 | 17, 25, 27 | syl2anc 667 |
. . . . . 6
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29 | 24, 28 | eleqtrrd 2532 |
. . . . 5
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30 | elmapfn 7494 |
. . . . 5
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31 | 29, 30 | syl 17 |
. . . 4
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32 | df-ov 6293 |
. . . . . 6
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33 | 32 | eqcomi 2460 |
. . . . 5
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34 | 33 | a1i 11 |
. . . 4
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35 | 19, 20, 31, 34 | ofc1 6554 |
. . 3
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36 | 13, 35 | mpdan 674 |
. 2
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37 | 9, 11, 36 | 3eqtrd 2489 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-ot 3977 df-uni 4199 df-int 4235 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-of 6531 df-om 6693 df-1st 6793 df-2nd 6794 df-supp 6915 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-map 7474 df-ixp 7523 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-fsupp 7884 df-sup 7956 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-nn 10610 df-2 10668 df-3 10669 df-4 10670 df-5 10671 df-6 10672 df-7 10673 df-8 10674 df-9 10675 df-10 10676 df-n0 10870 df-z 10938 df-dec 11052 df-uz 11160 df-fz 11785 df-struct 15123 df-ndx 15124 df-slot 15125 df-base 15126 df-sets 15127 df-ress 15128 df-plusg 15203 df-mulr 15204 df-sca 15206 df-vsca 15207 df-ip 15208 df-tset 15209 df-ple 15210 df-ds 15212 df-hom 15214 df-cco 15215 df-0g 15340 df-prds 15346 df-pws 15348 df-sra 18395 df-rgmod 18396 df-dsmm 19295 df-frlm 19310 df-mat 19433 |
This theorem is referenced by: dmatscmcl 19528 scmatscmide 19532 scmatscm 19538 mat2pmatlin 19759 monmatcollpw 19803 pmatcollpwlem 19804 chpmat1dlem 19859 chpdmatlem2 19863 chpdmatlem3 19864 |
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