![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pj1eq | Structured version Visualization version GIF version |
Description: Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pj1eu.a | โข + = (+gโ๐บ) |
pj1eu.s | โข โ = (LSSumโ๐บ) |
pj1eu.o | โข 0 = (0gโ๐บ) |
pj1eu.z | โข ๐ = (Cntzโ๐บ) |
pj1eu.2 | โข (๐ โ ๐ โ (SubGrpโ๐บ)) |
pj1eu.3 | โข (๐ โ ๐ โ (SubGrpโ๐บ)) |
pj1eu.4 | โข (๐ โ (๐ โฉ ๐) = { 0 }) |
pj1eu.5 | โข (๐ โ ๐ โ (๐โ๐)) |
pj1f.p | โข ๐ = (proj1โ๐บ) |
pj1eq.5 | โข (๐ โ ๐ โ (๐ โ ๐)) |
pj1eq.6 | โข (๐ โ ๐ต โ ๐) |
pj1eq.7 | โข (๐ โ ๐ถ โ ๐) |
Ref | Expression |
---|---|
pj1eq | โข (๐ โ (๐ = (๐ต + ๐ถ) โ (((๐๐๐)โ๐) = ๐ต โง ((๐๐๐)โ๐) = ๐ถ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pj1eq.5 | . . . 4 โข (๐ โ ๐ โ (๐ โ ๐)) | |
2 | pj1eu.a | . . . . 5 โข + = (+gโ๐บ) | |
3 | pj1eu.s | . . . . 5 โข โ = (LSSumโ๐บ) | |
4 | pj1eu.o | . . . . 5 โข 0 = (0gโ๐บ) | |
5 | pj1eu.z | . . . . 5 โข ๐ = (Cntzโ๐บ) | |
6 | pj1eu.2 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐บ)) | |
7 | pj1eu.3 | . . . . 5 โข (๐ โ ๐ โ (SubGrpโ๐บ)) | |
8 | pj1eu.4 | . . . . 5 โข (๐ โ (๐ โฉ ๐) = { 0 }) | |
9 | pj1eu.5 | . . . . 5 โข (๐ โ ๐ โ (๐โ๐)) | |
10 | pj1f.p | . . . . 5 โข ๐ = (proj1โ๐บ) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj1id 19653 | . . . 4 โข ((๐ โง ๐ โ (๐ โ ๐)) โ ๐ = (((๐๐๐)โ๐) + ((๐๐๐)โ๐))) |
12 | 1, 11 | mpdan 686 | . . 3 โข (๐ โ ๐ = (((๐๐๐)โ๐) + ((๐๐๐)โ๐))) |
13 | 12 | eqeq1d 2730 | . 2 โข (๐ โ (๐ = (๐ต + ๐ถ) โ (((๐๐๐)โ๐) + ((๐๐๐)โ๐)) = (๐ต + ๐ถ))) |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj1f 19651 | . . . 4 โข (๐ โ (๐๐๐):(๐ โ ๐)โถ๐) |
15 | 14, 1 | ffvelcdmd 7095 | . . 3 โข (๐ โ ((๐๐๐)โ๐) โ ๐) |
16 | pj1eq.6 | . . 3 โข (๐ โ ๐ต โ ๐) | |
17 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | pj2f 19652 | . . . 4 โข (๐ โ (๐๐๐):(๐ โ ๐)โถ๐) |
18 | 17, 1 | ffvelcdmd 7095 | . . 3 โข (๐ โ ((๐๐๐)โ๐) โ ๐) |
19 | pj1eq.7 | . . 3 โข (๐ โ ๐ถ โ ๐) | |
20 | 2, 4, 5, 6, 7, 8, 9, 15, 16, 18, 19 | subgdisjb 19647 | . 2 โข (๐ โ ((((๐๐๐)โ๐) + ((๐๐๐)โ๐)) = (๐ต + ๐ถ) โ (((๐๐๐)โ๐) = ๐ต โง ((๐๐๐)โ๐) = ๐ถ))) |
21 | 13, 20 | bitrd 279 | 1 โข (๐ โ (๐ = (๐ต + ๐ถ) โ (((๐๐๐)โ๐) = ๐ต โง ((๐๐๐)โ๐) = ๐ถ))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wb 205 โง wa 395 = wceq 1534 โ wcel 2099 โฉ cin 3946 โ wss 3947 {csn 4629 โcfv 6548 (class class class)co 7420 +gcplusg 17232 0gc0g 17420 SubGrpcsubg 19074 Cntzccntz 19265 LSSumclsm 19588 proj1cpj1 19589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cntz 19267 df-lsm 19590 df-pj1 19591 |
This theorem is referenced by: pj1lid 19655 pj1rid 19656 pj1ghm 19657 lsmhash 19659 dpjidcl 20014 pj1lmhm 20984 |
Copyright terms: Public domain | W3C validator |