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Mirrors > Home > MPE Home > Th. List > subgdisj2 | Structured version Visualization version GIF version |
Description: Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
Ref | Expression |
---|---|
subgdisj.p | โข + = (+gโ๐บ) |
subgdisj.o | โข 0 = (0gโ๐บ) |
subgdisj.z | โข ๐ = (Cntzโ๐บ) |
subgdisj.t | โข (๐ โ ๐ โ (SubGrpโ๐บ)) |
subgdisj.u | โข (๐ โ ๐ โ (SubGrpโ๐บ)) |
subgdisj.i | โข (๐ โ (๐ โฉ ๐) = { 0 }) |
subgdisj.s | โข (๐ โ ๐ โ (๐โ๐)) |
subgdisj.a | โข (๐ โ ๐ด โ ๐) |
subgdisj.c | โข (๐ โ ๐ถ โ ๐) |
subgdisj.b | โข (๐ โ ๐ต โ ๐) |
subgdisj.d | โข (๐ โ ๐ท โ ๐) |
subgdisj.j | โข (๐ โ (๐ด + ๐ต) = (๐ถ + ๐ท)) |
Ref | Expression |
---|---|
subgdisj2 | โข (๐ โ ๐ต = ๐ท) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgdisj.p | . 2 โข + = (+gโ๐บ) | |
2 | subgdisj.o | . 2 โข 0 = (0gโ๐บ) | |
3 | subgdisj.z | . 2 โข ๐ = (Cntzโ๐บ) | |
4 | subgdisj.u | . 2 โข (๐ โ ๐ โ (SubGrpโ๐บ)) | |
5 | subgdisj.t | . 2 โข (๐ โ ๐ โ (SubGrpโ๐บ)) | |
6 | incom 4148 | . . 3 โข (๐ โฉ ๐) = (๐ โฉ ๐) | |
7 | subgdisj.i | . . 3 โข (๐ โ (๐ โฉ ๐) = { 0 }) | |
8 | 6, 7 | eqtr3id 2790 | . 2 โข (๐ โ (๐ โฉ ๐) = { 0 }) |
9 | subgdisj.s | . . 3 โข (๐ โ ๐ โ (๐โ๐)) | |
10 | 3, 5, 4, 9 | cntzrecd 19379 | . 2 โข (๐ โ ๐ โ (๐โ๐)) |
11 | subgdisj.b | . 2 โข (๐ โ ๐ต โ ๐) | |
12 | subgdisj.d | . 2 โข (๐ โ ๐ท โ ๐) | |
13 | subgdisj.a | . 2 โข (๐ โ ๐ด โ ๐) | |
14 | subgdisj.c | . 2 โข (๐ โ ๐ถ โ ๐) | |
15 | subgdisj.j | . . 3 โข (๐ โ (๐ด + ๐ต) = (๐ถ + ๐ท)) | |
16 | 9, 13 | sseldd 3933 | . . . 4 โข (๐ โ ๐ด โ (๐โ๐)) |
17 | 1, 3 | cntzi 19031 | . . . 4 โข ((๐ด โ (๐โ๐) โง ๐ต โ ๐) โ (๐ด + ๐ต) = (๐ต + ๐ด)) |
18 | 16, 11, 17 | syl2anc 584 | . . 3 โข (๐ โ (๐ด + ๐ต) = (๐ต + ๐ด)) |
19 | 9, 14 | sseldd 3933 | . . . 4 โข (๐ โ ๐ถ โ (๐โ๐)) |
20 | 1, 3 | cntzi 19031 | . . . 4 โข ((๐ถ โ (๐โ๐) โง ๐ท โ ๐) โ (๐ถ + ๐ท) = (๐ท + ๐ถ)) |
21 | 19, 12, 20 | syl2anc 584 | . . 3 โข (๐ โ (๐ถ + ๐ท) = (๐ท + ๐ถ)) |
22 | 15, 18, 21 | 3eqtr3d 2784 | . 2 โข (๐ โ (๐ต + ๐ด) = (๐ท + ๐ถ)) |
23 | 1, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22 | subgdisj1 19392 | 1 โข (๐ โ ๐ต = ๐ท) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1540 โ wcel 2105 โฉ cin 3897 โ wss 3898 {csn 4573 โcfv 6479 (class class class)co 7337 +gcplusg 17059 0gc0g 17247 SubGrpcsubg 18845 Cntzccntz 19017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-0g 17249 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-grp 18676 df-minusg 18677 df-sbg 18678 df-subg 18848 df-cntz 19019 |
This theorem is referenced by: subgdisjb 19394 lvecindp 20506 lshpsmreu 37384 lshpkrlem5 37389 |
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