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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 | ⊢ + = (+g‘𝐺) |
subgdisj.o | ⊢ 0 = (0g‘𝐺) |
subgdisj.z | ⊢ 𝑍 = (Cntz‘𝐺) |
subgdisj.t | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
subgdisj.u | ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
subgdisj.i | ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) |
subgdisj.s | ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) |
subgdisj.a | ⊢ (𝜑 → 𝐴 ∈ 𝑇) |
subgdisj.c | ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
subgdisj.b | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
subgdisj.d | ⊢ (𝜑 → 𝐷 ∈ 𝑈) |
subgdisj.j | ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) |
Ref | Expression |
---|---|
subgdisj2 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subgdisj.p | . 2 ⊢ + = (+g‘𝐺) | |
2 | subgdisj.o | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | subgdisj.z | . 2 ⊢ 𝑍 = (Cntz‘𝐺) | |
4 | subgdisj.u | . 2 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) | |
5 | subgdisj.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
6 | incom 4175 | . . 3 ⊢ (𝑇 ∩ 𝑈) = (𝑈 ∩ 𝑇) | |
7 | subgdisj.i | . . 3 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) | |
8 | 6, 7 | syl5eqr 2867 | . 2 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 }) |
9 | subgdisj.s | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) | |
10 | 3, 5, 4, 9 | cntzrecd 18733 | . 2 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇)) |
11 | subgdisj.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
12 | subgdisj.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑈) | |
13 | subgdisj.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
14 | subgdisj.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑇) | |
15 | subgdisj.j | . . 3 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
16 | 9, 13 | sseldd 3965 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑍‘𝑈)) |
17 | 1, 3 | cntzi 18397 | . . . 4 ⊢ ((𝐴 ∈ (𝑍‘𝑈) ∧ 𝐵 ∈ 𝑈) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
18 | 16, 11, 17 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
19 | 9, 14 | sseldd 3965 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝑍‘𝑈)) |
20 | 1, 3 | cntzi 18397 | . . . 4 ⊢ ((𝐶 ∈ (𝑍‘𝑈) ∧ 𝐷 ∈ 𝑈) → (𝐶 + 𝐷) = (𝐷 + 𝐶)) |
21 | 19, 12, 20 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) = (𝐷 + 𝐶)) |
22 | 15, 18, 21 | 3eqtr3d 2861 | . 2 ⊢ (𝜑 → (𝐵 + 𝐴) = (𝐷 + 𝐶)) |
23 | 1, 2, 3, 4, 5, 8, 10, 11, 12, 13, 14, 22 | subgdisj1 18746 | 1 ⊢ (𝜑 → 𝐵 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 {csn 4557 ‘cfv 6348 (class class class)co 7145 +gcplusg 16553 0gc0g 16701 SubGrpcsubg 18211 Cntzccntz 18383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 |
This theorem is referenced by: subgdisjb 18748 lvecindp 19839 lshpsmreu 36125 lshpkrlem5 36130 |
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