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 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
nmzsubg.2	⊢ 𝑋 = (Base‘𝐺)
nmzsubg.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ssnmz	⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)

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.

Step	Hyp	Ref	Expression
1		nmzsubg.2	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
2	1	subgss 12965	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
3	2	sselda 3155	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
4		simpll 527	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
5		subgrcl 12970	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝐺 ∈ Grp)
7	4, 2	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆 ⊆ 𝑋)
8		simplrl 535	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧 ∈ 𝑆)
9	7, 8	sseldd 3156	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧 ∈ 𝑋)
10		nmzsubg.3	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝐺)
11		eqid 2177	. . . . . . . . . . . . 13 ⊢ (0g‘𝐺) = (0g‘𝐺)
12		eqid 2177	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
13	1, 10, 11, 12	grplinv 12854	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
14	6, 9, 13	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
15	14	oveq1d 5887	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g‘𝐺) + 𝑤))
16	12	subginvcl 12974	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → ((invg‘𝐺)‘𝑧) ∈ 𝑆)
17	4, 8, 16	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg‘𝐺)‘𝑧) ∈ 𝑆)
18	7, 17	sseldd 3156	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
19		simplrr 536	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤 ∈ 𝑋)
20	1, 10	grpass 12818	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑧) ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
21	6, 18, 9, 19, 20	syl13anc 1240	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)))
22	1, 10, 11	grplid 12838	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
23	6, 19, 22	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((0g‘𝐺) + 𝑤) = 𝑤)
24	15, 21, 23	3eqtr3d 2218	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) = 𝑤)
25		simpr 110	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
26	10	subgcl 12975	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑆 ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
27	4, 17, 25, 26	syl3anc 1238	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
28	24, 27	eqeltrrd 2255	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤 ∈ 𝑆)
29	10	subgcl 12975	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑤 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
30	4, 28, 8, 29	syl3anc 1238	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
31		simpll 527	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
32		simplrl 535	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧 ∈ 𝑆)
33	31, 5	syl 14	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝐺 ∈ Grp)
34		simplrr 536	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤 ∈ 𝑋)
35	31, 32, 3	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧 ∈ 𝑋)
36		eqid 2177	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
37	1, 10, 36	grppncan 12893	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) = 𝑤)
38	33, 34, 35, 37	syl3anc 1238	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) = 𝑤)
39		simpr 110	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
40	36	subgsubcl 12976	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑤 + 𝑧) ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) ∈ 𝑆)
41	31, 39, 32, 40	syl3anc 1238	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) ∈ 𝑆)
42	38, 41	eqeltrrd 2255	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤 ∈ 𝑆)
43	10	subgcl 12975	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
44	31, 32, 42, 43	syl3anc 1238	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
45	30, 44	impbida 596	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
46	45	anassrs 400	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ 𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
47	46	ralrimiva 2550	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
48		elnmz.1	. . . . 5 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)}
49	48	elnmz 12999	. . . 4 ⊢ (𝑧 ∈ 𝑁 ↔ (𝑧 ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
50	3, 47, 49	sylanbrc 417	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑁)
51	50	ex 115	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 ∈ 𝑆 → 𝑧 ∈ 𝑁))
52	51	ssrdv 3161	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)

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  0_gc0g 12693  Grpcgrp 12809  inv_gcminusg 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