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Mirrors > Home > MPE Home > Th. List > pjff | Structured version Unicode version |
Description: A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjf.k |
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Ref | Expression |
---|---|
pjff |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2454 |
. . . 4
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2 | eqid 2454 |
. . . 4
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3 | eqid 2454 |
. . . 4
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4 | eqid 2454 |
. . . 4
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5 | phllmod 18185 |
. . . . 5
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6 | 5 | adantr 465 |
. . . 4
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7 | eqid 2454 |
. . . . . 6
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8 | eqid 2454 |
. . . . . 6
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9 | pjf.k |
. . . . . 6
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10 | 7, 1, 8, 2, 9 | pjdm2 18262 |
. . . . 5
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11 | 10 | simprbda 623 |
. . . 4
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12 | 7, 1 | lssss 17142 |
. . . . . 6
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13 | 11, 12 | syl 16 |
. . . . 5
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14 | 7, 8, 1 | ocvlss 18223 |
. . . . 5
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15 | 13, 14 | syldan 470 |
. . . 4
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16 | 8, 1, 3 | ocvin 18225 |
. . . . 5
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17 | 11, 16 | syldan 470 |
. . . 4
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18 | 1, 2, 3, 4, 6, 11, 15, 17 | pj1lmhm 17305 |
. . 3
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19 | 10 | simplbda 624 |
. . . . . 6
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20 | 19 | oveq2d 6217 |
. . . . 5
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21 | 7 | ressid 14353 |
. . . . . 6
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22 | 21 | adantr 465 |
. . . . 5
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23 | 20, 22 | eqtrd 2495 |
. . . 4
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24 | 23 | oveq1d 6216 |
. . 3
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25 | 18, 24 | eleqtrd 2544 |
. 2
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26 | 8, 4, 9 | pjfval2 18260 |
. 2
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27 | 25, 26 | fmptd 5977 |
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 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-pre-mulgt0 9471 |
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-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-riota 6162 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-om 6588 df-1st 6688 df-2nd 6689 df-recs 6943 df-rdg 6977 df-er 7212 df-map 7327 df-en 7422 df-dom 7423 df-sdom 7424 df-pnf 9532 df-mnf 9533 df-xr 9534 df-ltxr 9535 df-le 9536 df-sub 9709 df-neg 9710 df-nn 10435 df-2 10492 df-3 10493 df-4 10494 df-5 10495 df-6 10496 df-7 10497 df-8 10498 df-ndx 14296 df-slot 14297 df-base 14298 df-sets 14299 df-ress 14300 df-plusg 14371 df-sca 14374 df-vsca 14375 df-ip 14376 df-0g 14500 df-mnd 15535 df-submnd 15585 df-grp 15665 df-minusg 15666 df-sbg 15667 df-subg 15798 df-ghm 15865 df-cntz 15955 df-lsm 16257 df-pj1 16258 df-cmn 16401 df-abl 16402 df-mgp 16715 df-ur 16727 df-rng 16771 df-lmod 17074 df-lss 17138 df-lmhm 17227 df-lvec 17308 df-sra 17377 df-rgmod 17378 df-phl 18181 df-ocv 18214 df-pj 18254 |
This theorem is referenced by: (None) |
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