Theorem hhssabloilem 28118

Description: Lemma for hhssabloi 28119. Formerly part of proof for hhssabloi 28119 which was based on the deprecated definition "SubGrpOp" for subgroups. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 27-Aug-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
hhssabl.1	|- H e. SH

Assertion

Ref	Expression
hhssabloilem	|- ( +h e. GrpOp /\ ( +h |` ( H X. H ) ) e. GrpOp /\ ( +h |` ( H X. H ) ) C_ +h )

Proof of Theorem hhssabloilem

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hilablo 28017	. . 3 |- +h e. AbelOp
2		ablogrpo 27401	. . 3 |- ( +h e. AbelOp -> +h e. GrpOp )
3	1, 2	ax-mp 5	. 2 |- +h e. GrpOp
4		hhssabl.1	. . . 4 |- H e. SH
5	4	elexi 3213	. . 3 |- H e. _V
6		eqid 2622	. . . . . . . 8 |- ran +h = ran +h
7	6	grpofo 27353	. . . . . . 7 |- ( +h e. GrpOp -> +h : ( ran +h X. ran +h ) -onto-> ran +h )
8		fof 6115	. . . . . . 7 |- ( +h : ( ran +h X. ran +h ) -onto-> ran +h -> +h : ( ran +h X. ran +h ) --> ran +h )
9	3, 7, 8	mp2b 10	. . . . . 6 |- +h : ( ran +h X. ran +h ) --> ran +h
10	4	shssii 28070	. . . . . . . 8 |- H C_ ~H
11		df-hba 27826	. . . . . . . . 9 |- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
12		eqid 2622	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
13	12	hhva 28023	. . . . . . . . 9 |- +h = ( +v ` <. <. +h , .h >. , normh >. )
14	11, 13	bafval 27459	. . . . . . . 8 |- ~H = ran +h
15	10, 14	sseqtri 3637	. . . . . . 7 |- H C_ ran +h
16		xpss12 5225	. . . . . . 7 |- ( ( H C_ ran +h /\ H C_ ran +h ) -> ( H X. H ) C_ ( ran +h X. ran +h ) )
17	15, 15, 16	mp2an 708	. . . . . 6 |- ( H X. H ) C_ ( ran +h X. ran +h )
18		fssres 6070	. . . . . 6 |- ( ( +h : ( ran +h X. ran +h ) --> ran +h /\ ( H X. H ) C_ ( ran +h X. ran +h ) ) -> ( +h |` ( H X. H ) ) : ( H X. H ) --> ran +h )
19	9, 17, 18	mp2an 708	. . . . 5 |- ( +h |` ( H X. H ) ) : ( H X. H ) --> ran +h
20		ffn 6045	. . . . 5 |- ( ( +h |` ( H X. H ) ) : ( H X. H ) --> ran +h -> ( +h |` ( H X. H ) ) Fn ( H X. H ) )
21	19, 20	ax-mp 5	. . . 4 |- ( +h |` ( H X. H ) ) Fn ( H X. H )
22		ovres 6800	. . . . . 6 |- ( ( x e. H /\ y e. H ) -> ( x ( +h |` ( H X. H ) ) y ) = ( x +h y ) )
23		shaddcl 28074	. . . . . . 7 |- ( ( H e. SH /\ x e. H /\ y e. H ) -> ( x +h y ) e. H )
24	4, 23	mp3an1 1411	. . . . . 6 |- ( ( x e. H /\ y e. H ) -> ( x +h y ) e. H )
25	22, 24	eqeltrd 2701	. . . . 5 |- ( ( x e. H /\ y e. H ) -> ( x ( +h |` ( H X. H ) ) y ) e. H )
26	25	rgen2a 2977	. . . 4 |- A. x e. H A. y e. H ( x ( +h |` ( H X. H ) ) y ) e. H
27		ffnov 6764	. . . 4 |- ( ( +h |` ( H X. H ) ) : ( H X. H ) --> H <-> ( ( +h |` ( H X. H ) ) Fn ( H X. H ) /\ A. x e. H A. y e. H ( x ( +h |` ( H X. H ) ) y ) e. H ) )
28	21, 26, 27	mpbir2an 955	. . 3 |- ( +h |` ( H X. H ) ) : ( H X. H ) --> H
29	22	oveq1d 6665	. . . . 5 |- ( ( x e. H /\ y e. H ) -> ( ( x ( +h |` ( H X. H ) ) y ) +h z ) = ( ( x +h y ) +h z ) )
30	29	3adant3 1081	. . . 4 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( ( x ( +h |` ( H X. H ) ) y ) +h z ) = ( ( x +h y ) +h z ) )
31		ovres 6800	. . . . 5 |- ( ( ( x ( +h |` ( H X. H ) ) y ) e. H /\ z e. H ) -> ( ( x ( +h |` ( H X. H ) ) y ) ( +h |` ( H X. H ) ) z ) = ( ( x ( +h |` ( H X. H ) ) y ) +h z ) )
32	25, 31	stoic3 1701	. . . 4 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( ( x ( +h |` ( H X. H ) ) y ) ( +h |` ( H X. H ) ) z ) = ( ( x ( +h |` ( H X. H ) ) y ) +h z ) )
33		ovres 6800	. . . . . . 7 |- ( ( y e. H /\ z e. H ) -> ( y ( +h |` ( H X. H ) ) z ) = ( y +h z ) )
34	33	oveq2d 6666	. . . . . 6 |- ( ( y e. H /\ z e. H ) -> ( x +h ( y ( +h |` ( H X. H ) ) z ) ) = ( x +h ( y +h z ) ) )
35	34	3adant1 1079	. . . . 5 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( x +h ( y ( +h |` ( H X. H ) ) z ) ) = ( x +h ( y +h z ) ) )
36	28	fovcl 6765	. . . . . . 7 |- ( ( y e. H /\ z e. H ) -> ( y ( +h |` ( H X. H ) ) z ) e. H )
37		ovres 6800	. . . . . . 7 |- ( ( x e. H /\ ( y ( +h |` ( H X. H ) ) z ) e. H ) -> ( x ( +h |` ( H X. H ) ) ( y ( +h |` ( H X. H ) ) z ) ) = ( x +h ( y ( +h |` ( H X. H ) ) z ) ) )
38	36, 37	sylan2 491	. . . . . 6 |- ( ( x e. H /\ ( y e. H /\ z e. H ) ) -> ( x ( +h |` ( H X. H ) ) ( y ( +h |` ( H X. H ) ) z ) ) = ( x +h ( y ( +h |` ( H X. H ) ) z ) ) )
39	38	3impb 1260	. . . . 5 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( x ( +h |` ( H X. H ) ) ( y ( +h |` ( H X. H ) ) z ) ) = ( x +h ( y ( +h |` ( H X. H ) ) z ) ) )
40	15	sseli 3599	. . . . . 6 |- ( x e. H -> x e. ran +h )
41	15	sseli 3599	. . . . . 6 |- ( y e. H -> y e. ran +h )
42	15	sseli 3599	. . . . . 6 |- ( z e. H -> z e. ran +h )
43	6	grpoass 27357	. . . . . . 7 |- ( ( +h e. GrpOp /\ ( x e. ran +h /\ y e. ran +h /\ z e. ran +h ) ) -> ( ( x +h y ) +h z ) = ( x +h ( y +h z ) ) )
44	3, 43	mpan 706	. . . . . 6 |- ( ( x e. ran +h /\ y e. ran +h /\ z e. ran +h ) -> ( ( x +h y ) +h z ) = ( x +h ( y +h z ) ) )
45	40, 41, 42, 44	syl3an 1368	. . . . 5 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( ( x +h y ) +h z ) = ( x +h ( y +h z ) ) )
46	35, 39, 45	3eqtr4d 2666	. . . 4 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( x ( +h |` ( H X. H ) ) ( y ( +h |` ( H X. H ) ) z ) ) = ( ( x +h y ) +h z ) )
47	30, 32, 46	3eqtr4d 2666	. . 3 |- ( ( x e. H /\ y e. H /\ z e. H ) -> ( ( x ( +h |` ( H X. H ) ) y ) ( +h |` ( H X. H ) ) z ) = ( x ( +h |` ( H X. H ) ) ( y ( +h |` ( H X. H ) ) z ) ) )
48		hilid 28018	. . . 4 |- (GId ` +h ) = 0h
49		sh0 28073	. . . . 5 |- ( H e. SH -> 0h e. H )
50	4, 49	ax-mp 5	. . . 4 |- 0h e. H
51	48, 50	eqeltri 2697	. . 3 |- (GId ` +h ) e. H
52		ovres 6800	. . . . 5 |- ( ( (GId ` +h ) e. H /\ x e. H ) -> ( (GId ` +h ) ( +h |` ( H X. H ) ) x ) = ( (GId ` +h ) +h x ) )
53	51, 52	mpan 706	. . . 4 |- ( x e. H -> ( (GId ` +h ) ( +h |` ( H X. H ) ) x ) = ( (GId ` +h ) +h x ) )
54		eqid 2622	. . . . . 6 |- (GId ` +h ) = (GId ` +h )
55	6, 54	grpolid 27370	. . . . 5 |- ( ( +h e. GrpOp /\ x e. ran +h ) -> ( (GId ` +h ) +h x ) = x )
56	3, 40, 55	sylancr 695	. . . 4 |- ( x e. H -> ( (GId ` +h ) +h x ) = x )
57	53, 56	eqtrd 2656	. . 3 |- ( x e. H -> ( (GId ` +h ) ( +h |` ( H X. H ) ) x ) = x )
58	12	hhnv 28022	. . . . . . 7 |- <. <. +h , .h >. , normh >. e. NrmCVec
59	12	hhsm 28026	. . . . . . . 8 |- .h = ( .sOLD ` <. <. +h , .h >. , normh >. )
60		eqid 2622	. . . . . . . 8 |- ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) = ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) )
61	13, 59, 60	nvinvfval 27495	. . . . . . 7 |- ( <. <. +h , .h >. , normh >. e. NrmCVec -> ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) = ( inv ` +h ) )
62	58, 61	ax-mp 5	. . . . . 6 |- ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) = ( inv ` +h )
63	62	eqcomi 2631	. . . . 5 |- ( inv ` +h ) = ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) )
64	63	fveq1i 6192	. . . 4 |- ( ( inv ` +h ) ` x ) = ( ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) ` x )
65		ax-hfvmul 27862	. . . . . . 7 |- .h : ( CC X. ~H ) --> ~H
66		ffn 6045	. . . . . . 7 |- ( .h : ( CC X. ~H ) --> ~H -> .h Fn ( CC X. ~H ) )
67	65, 66	ax-mp 5	. . . . . 6 |- .h Fn ( CC X. ~H )
68		neg1cn 11124	. . . . . 6 |- -u 1 e. CC
69	60	curry1val 7270	. . . . . 6 |- ( ( .h Fn ( CC X. ~H ) /\ -u 1 e. CC ) -> ( ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) ` x ) = ( -u 1 .h x ) )
70	67, 68, 69	mp2an 708	. . . . 5 |- ( ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) ` x ) = ( -u 1 .h x )
71		shmulcl 28075	. . . . . 6 |- ( ( H e. SH /\ -u 1 e. CC /\ x e. H ) -> ( -u 1 .h x ) e. H )
72	4, 68, 71	mp3an12 1414	. . . . 5 |- ( x e. H -> ( -u 1 .h x ) e. H )
73	70, 72	syl5eqel 2705	. . . 4 |- ( x e. H -> ( ( .h o. `' ( 2nd |` ( { -u 1 } X. _V ) ) ) ` x ) e. H )
74	64, 73	syl5eqel 2705	. . 3 |- ( x e. H -> ( ( inv ` +h ) ` x ) e. H )
75		ovres 6800	. . . . 5 |- ( ( ( ( inv ` +h ) ` x ) e. H /\ x e. H ) -> ( ( ( inv ` +h ) ` x ) ( +h |` ( H X. H ) ) x ) = ( ( ( inv ` +h ) ` x ) +h x ) )
76	74, 75	mpancom 703	. . . 4 |- ( x e. H -> ( ( ( inv ` +h ) ` x ) ( +h |` ( H X. H ) ) x ) = ( ( ( inv ` +h ) ` x ) +h x ) )
77		eqid 2622	. . . . . 6 |- ( inv ` +h ) = ( inv ` +h )
78	6, 54, 77	grpolinv 27380	. . . . 5 |- ( ( +h e. GrpOp /\ x e. ran +h ) -> ( ( ( inv ` +h ) ` x ) +h x ) = (GId ` +h ) )
79	3, 40, 78	sylancr 695	. . . 4 |- ( x e. H -> ( ( ( inv ` +h ) ` x ) +h x ) = (GId ` +h ) )
80	76, 79	eqtrd 2656	. . 3 |- ( x e. H -> ( ( ( inv ` +h ) ` x ) ( +h |` ( H X. H ) ) x ) = (GId ` +h ) )
81	5, 28, 47, 51, 57, 74, 80	isgrpoi 27352	. 2 |- ( +h |` ( H X. H ) ) e. GrpOp
82		resss 5422	. 2 |- ( +h |` ( H X. H ) ) C_ +h
83	3, 81, 82	3pm3.2i 1239	1 |- ( +h e. GrpOp /\ ( +h |` ( H X. H ) ) e. GrpOp /\ ( +h |` ( H X. H ) ) C_ +h )

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   2ndc2nd 7167   CCcc 9934   1c1 9937   -ucneg 10267   GrpOpcgr 27343  GIdcgi 27344   invcgn 27345   AbelOpcablo 27398   NrmCVeccnv 27439   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780   0hc0v 27781   SHcsh 27785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-sh 28064

This theorem is referenced by:  hhssabloi  28119