![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > pjcss | Structured version Unicode version |
Description: A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjcss.k |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
pjcss.c |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
pjcss |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjcss.c |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | eqid 2454 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | eqid 2454 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | eqid 2454 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | simpl 457 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | eqid 2454 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | pjcss.k |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 2, 6, 3, 4, 7 | pjdm2 18264 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8 | simprbda 623 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 2, 6 | lssss 17144 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 9, 10 | syl 16 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 2, 3 | ocvss 18223 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 8 | simplbda 624 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 13 | syl5sseqr 3516 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 1, 2, 3, 4, 5, 11, 14 | lsmcss 18245 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 15 | ex 434 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | ssrdv 3473 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-rmo 2807 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-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-tpos 6858 df-recs 6945 df-rdg 6979 df-er 7214 df-map 7329 df-en 7424 df-dom 7425 df-sdom 7426 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-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-0g 14502 df-mnd 15537 df-mhm 15586 df-grp 15667 df-minusg 15668 df-sbg 15669 df-subg 15800 df-ghm 15867 df-cntz 15957 df-lsm 16259 df-pj1 16260 df-cmn 16403 df-abl 16404 df-mgp 16717 df-ur 16729 df-rng 16773 df-oppr 16841 df-rnghom 16932 df-staf 17056 df-srng 17057 df-lmod 17076 df-lss 17140 df-lmhm 17229 df-lvec 17310 df-sra 17379 df-rgmod 17380 df-phl 18183 df-ocv 18216 df-css 18217 df-pj 18256 |
This theorem is referenced by: ocvpj 18270 ishil2 18272 cldcss 21063 hlhil 21065 |
Copyright terms: Public domain | W3C validator |