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Theorem ssnmz 13002
Description: A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
elnmz.1 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
nmzsubg.2 𝑋 = (Base‘𝐺)
nmzsubg.3 + = (+g𝐺)
Assertion
Ref Expression
ssnmz (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem ssnmz
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmzsubg.2 . . . . . 6 𝑋 = (Base‘𝐺)
21subgss 12965 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑋)
32sselda 3155 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → 𝑧𝑋)
4 simpll 527 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
5 subgrcl 12970 . . . . . . . . . . . . 13 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
64, 5syl 14 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝐺 ∈ Grp)
74, 2syl 14 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆𝑋)
8 simplrl 535 . . . . . . . . . . . . 13 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧𝑆)
97, 8sseldd 3156 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧𝑋)
10 nmzsubg.3 . . . . . . . . . . . . 13 + = (+g𝐺)
11 eqid 2177 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
12 eqid 2177 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
131, 10, 11, 12grplinv 12854 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑧𝑋) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
146, 9, 13syl2anc 411 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + 𝑧) = (0g𝐺))
1514oveq1d 5887 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g𝐺) + 𝑤))
1612subginvcl 12974 . . . . . . . . . . . . 13 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → ((invg𝐺)‘𝑧) ∈ 𝑆)
174, 8, 16syl2anc 411 . . . . . . . . . . . 12 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg𝐺)‘𝑧) ∈ 𝑆)
187, 17sseldd 3156 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg𝐺)‘𝑧) ∈ 𝑋)
19 simplrr 536 . . . . . . . . . . 11 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤𝑋)
201, 10grpass 12818 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑧) ∈ 𝑋𝑧𝑋𝑤𝑋)) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
216, 18, 9, 19, 20syl13anc 1240 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)))
221, 10, 11grplid 12838 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺) + 𝑤) = 𝑤)
236, 19, 22syl2anc 411 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((0g𝐺) + 𝑤) = 𝑤)
2415, 21, 233eqtr3d 2218 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
25 simpr 110 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
2610subgcl 12975 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((invg𝐺)‘𝑧) ∈ 𝑆 ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
274, 17, 25, 26syl3anc 1238 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
2824, 27eqeltrrd 2255 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤𝑆)
2910subgcl 12975 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑤𝑆𝑧𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
304, 28, 8, 29syl3anc 1238 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
31 simpll 527 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
32 simplrl 535 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧𝑆)
3331, 5syl 14 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝐺 ∈ Grp)
34 simplrr 536 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤𝑋)
3531, 32, 3syl2anc 411 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧𝑋)
36 eqid 2177 . . . . . . . . . . 11 (-g𝐺) = (-g𝐺)
371, 10, 36grppncan 12893 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑤𝑋𝑧𝑋) → ((𝑤 + 𝑧)(-g𝐺)𝑧) = 𝑤)
3833, 34, 35, 37syl3anc 1238 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g𝐺)𝑧) = 𝑤)
39 simpr 110 . . . . . . . . . 10 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
4036subgsubcl 12976 . . . . . . . . . 10 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑤 + 𝑧) ∈ 𝑆𝑧𝑆) → ((𝑤 + 𝑧)(-g𝐺)𝑧) ∈ 𝑆)
4131, 39, 32, 40syl3anc 1238 . . . . . . . . 9 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g𝐺)𝑧) ∈ 𝑆)
4238, 41eqeltrrd 2255 . . . . . . . 8 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤𝑆)
4310subgcl 12975 . . . . . . . 8 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆𝑤𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
4431, 32, 42, 43syl3anc 1238 . . . . . . 7 (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
4530, 44impbida 596 . . . . . 6 ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧𝑆𝑤𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
4645anassrs 400 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) ∧ 𝑤𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
4746ralrimiva 2550 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → ∀𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
48 elnmz.1 . . . . 5 𝑁 = {𝑥𝑋 ∣ ∀𝑦𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
4948elnmz 12999 . . . 4 (𝑧𝑁 ↔ (𝑧𝑋 ∧ ∀𝑤𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
503, 47, 49sylanbrc 417 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑆) → 𝑧𝑁)
5150ex 115 . 2 (𝑆 ∈ (SubGrp‘𝐺) → (𝑧𝑆𝑧𝑁))
5251ssrdv 3161 1 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝑁)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  {crab 2459  wss 3129  cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  0gc0g 12693  Grpcgrp 12809  invgcminusg 12810  -gcsg 12811  SubGrpcsubg 12958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-1st 6138  df-2nd 6139  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-iress 12462  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-sbg 12814  df-subg 12961
This theorem is referenced by:  nmznsg  13004
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