Theorem ssnmz 13991

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 13954	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
3	2	sselda 3248	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑋)
4		simpll 531	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
5		subgrcl 13959	. . . . . . . . . . . . 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 541	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧 ∈ 𝑆)
9	7, 8	sseldd 3249	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑧 ∈ 𝑋)
10		nmzsubg.3	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝐺)
11		eqid 2238	. . . . . . . . . . . . 13 ⊢ (0g‘𝐺) = (0g‘𝐺)
12		eqid 2238	. . . . . . . . . . . . 13 ⊢ (invg‘𝐺) = (invg‘𝐺)
13	1, 10, 11, 12	grplinv 13832	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑧 ∈ 𝑋) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
14	6, 9, 13	syl2anc 415	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + 𝑧) = (0g‘𝐺))
15	14	oveq1d 6090	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((((invg‘𝐺)‘𝑧) + 𝑧) + 𝑤) = ((0g‘𝐺) + 𝑤))
16	12	subginvcl 13963	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → ((invg‘𝐺)‘𝑧) ∈ 𝑆)
17	4, 8, 16	syl2anc 415	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg‘𝐺)‘𝑧) ∈ 𝑆)
18	7, 17	sseldd 3249	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((invg‘𝐺)‘𝑧) ∈ 𝑋)
19		simplrr 542	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤 ∈ 𝑋)
20	1, 10	grpass 13791	. . . . . . . . . . 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 1280	. . . . . . . . . 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 13813	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺) + 𝑤) = 𝑤)
23	6, 19, 22	syl2anc 415	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → ((0g‘𝐺) + 𝑤) = 𝑤)
24	15, 21, 23	3eqtr3d 2279	. . . . . . . . 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 13964	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ((invg‘𝐺)‘𝑧) ∈ 𝑆 ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
27	4, 17, 25, 26	syl3anc 1278	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (((invg‘𝐺)‘𝑧) + (𝑧 + 𝑤)) ∈ 𝑆)
28	24, 27	eqeltrrd 2316	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → 𝑤 ∈ 𝑆)
29	10	subgcl 13964	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑤 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
30	4, 28, 8, 29	syl3anc 1278	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑧 + 𝑤) ∈ 𝑆) → (𝑤 + 𝑧) ∈ 𝑆)
31		simpll 531	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑆 ∈ (SubGrp‘𝐺))
32		simplrl 541	. . . . . . . 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 542	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤 ∈ 𝑋)
35	31, 32, 3	syl2anc 415	. . . . . . . . . 10 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑧 ∈ 𝑋)
36		eqid 2238	. . . . . . . . . . 11 ⊢ (-g‘𝐺) = (-g‘𝐺)
37	1, 10, 36	grppncan 13873	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) = 𝑤)
38	33, 34, 35, 37	syl3anc 1278	. . . . . . . . 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 13965	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑤 + 𝑧) ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) ∈ 𝑆)
41	31, 39, 32, 40	syl3anc 1278	. . . . . . . . 9 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → ((𝑤 + 𝑧)(-g‘𝐺)𝑧) ∈ 𝑆)
42	38, 41	eqeltrrd 2316	. . . . . . . 8 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → 𝑤 ∈ 𝑆)
43	10	subgcl 13964	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
44	31, 32, 42, 43	syl3anc 1278	. . . . . . 7 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑤 + 𝑧) ∈ 𝑆) → (𝑧 + 𝑤) ∈ 𝑆)
45	30, 44	impbida 604	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑋)) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
46	45	anassrs 404	. . . . 5 ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) ∧ 𝑤 ∈ 𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))
47	46	ralrimiva 2623	. . . 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 13988	. . . 4 ⊢ (𝑧 ∈ 𝑁 ↔ (𝑧 ∈ 𝑋 ∧ ∀𝑤 ∈ 𝑋 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)))
50	3, 47, 49	sylanbrc 421	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑁)
51	50	ex 115	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑧 ∈ 𝑆 → 𝑧 ∈ 𝑁))
52	51	ssrdv 3254	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587  Grpcgrp 13782  inv_gcminusg 13783  -_gcsg 13784  SubGrpcsubg 13947

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-subg 13950

This theorem is referenced by:  nmznsg  13993