Theorem msubco 29158

Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypothesis

Ref	Expression
msubco.s	|- S = (mSubst ` T )

Assertion

Ref	Expression
msubco	|- ( ( F e. ran S /\ G e. ran S ) -> ( F o. G ) e. ran S )

Proof of Theorem msubco

Dummy variables f g h x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2454	. . . . 5 |- (mEx ` T ) = (mEx ` T )
2		eqid 2454	. . . . 5 |- (mRSubst ` T ) = (mRSubst ` T )
3		msubco.s	. . . . 5 |- S = (mSubst ` T )
4	1, 2, 3	elmsubrn 29155	. . . 4 |- ran S = ran ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) )
5	4	eleq2i 2532	. . 3 |- ( F e. ran S <-> F e. ran ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) ) )
6		eqid 2454	. . . 4 |- ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) ) = ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) )
7		fvex 5858	. . . . 5 |- (mEx ` T ) e. _V
8	7	mptex 6118	. . . 4 |- ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) e. _V
9	6, 8	elrnmpti 5242	. . 3 |- ( F e. ran ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) ) <-> E. f e. ran (mRSubst ` T ) F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) )
10	5, 9	bitri 249	. 2 |- ( F e. ran S <-> E. f e. ran (mRSubst ` T ) F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) )
11	1, 2, 3	elmsubrn 29155	. . . 4 |- ran S = ran ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
12	11	eleq2i 2532	. . 3 |- ( G e. ran S <-> G e. ran ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) )
13		eqid 2454	. . . 4 |- ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) = ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
14	7	mptex 6118	. . . 4 |- ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) e. _V
15	13, 14	elrnmpti 5242	. . 3 |- ( G e. ran ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) <-> E. g e. ran (mRSubst ` T ) G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
16	12, 15	bitri 249	. 2 |- ( G e. ran S <-> E. g e. ran (mRSubst ` T ) G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
17		reeanv 3022	. . 3 |- ( E. f e. ran (mRSubst ` T ) E. g e. ran (mRSubst ` T ) ( F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) <-> ( E. f e. ran (mRSubst ` T ) F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ E. g e. ran (mRSubst ` T ) G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) )
18		simpr 459	. . . . . . . . . . . 12 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> y e. (mEx ` T ) )
19		eqid 2454	. . . . . . . . . . . . 13 |- (mTC ` T ) = (mTC ` T )
20		eqid 2454	. . . . . . . . . . . . 13 |- (mREx ` T ) = (mREx ` T )
21	19, 1, 20	mexval 29129	. . . . . . . . . . . 12 |- (mEx ` T ) = ( (mTC ` T ) X. (mREx ` T ) )
22	18, 21	syl6eleq 2552	. . . . . . . . . . 11 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> y e. ( (mTC ` T ) X. (mREx ` T ) ) )
23		xp1st 6803	. . . . . . . . . . 11 |- ( y e. ( (mTC ` T ) X. (mREx ` T ) ) -> ( 1st ` y ) e. (mTC ` T ) )
24	22, 23	syl 16	. . . . . . . . . 10 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( 1st ` y ) e. (mTC ` T ) )
25	2, 20	mrsubf 29144	. . . . . . . . . . . 12 |- ( g e. ran (mRSubst ` T ) -> g : (mREx ` T ) --> (mREx ` T ) )
26	25	ad2antlr 724	. . . . . . . . . . 11 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> g : (mREx ` T ) --> (mREx ` T ) )
27		xp2nd 6804	. . . . . . . . . . . 12 |- ( y e. ( (mTC ` T ) X. (mREx ` T ) ) -> ( 2nd ` y ) e. (mREx ` T ) )
28	22, 27	syl 16	. . . . . . . . . . 11 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( 2nd ` y ) e. (mREx ` T ) )
29	26, 28	ffvelrnd 6008	. . . . . . . . . 10 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( g ` ( 2nd ` y ) ) e. (mREx ` T ) )
30		opelxpi 5020	. . . . . . . . . 10 |- ( ( ( 1st ` y ) e. (mTC ` T ) /\ ( g ` ( 2nd ` y ) ) e. (mREx ` T ) ) -> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. e. ( (mTC ` T ) X. (mREx ` T ) ) )
31	24, 29, 30	syl2anc 659	. . . . . . . . 9 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. e. ( (mTC ` T ) X. (mREx ` T ) ) )
32	31, 21	syl6eleqr 2553	. . . . . . . 8 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. e. (mEx ` T ) )
33		eqidd 2455	. . . . . . . 8 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
34		eqidd 2455	. . . . . . . 8 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) )
35		fvex 5858	. . . . . . . . . 10 |- ( 1st ` y ) e. _V
36		fvex 5858	. . . . . . . . . 10 |- ( g ` ( 2nd ` y ) ) e. _V
37	35, 36	op1std 6783	. . . . . . . . 9 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> ( 1st ` x ) = ( 1st ` y ) )
38	35, 36	op2ndd 6784	. . . . . . . . . 10 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> ( 2nd ` x ) = ( g ` ( 2nd ` y ) ) )
39	38	fveq2d 5852	. . . . . . . . 9 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> ( f ` ( 2nd ` x ) ) = ( f ` ( g ` ( 2nd ` y ) ) ) )
40	37, 39	opeq12d 4211	. . . . . . . 8 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. = <. ( 1st ` y ) , ( f ` ( g ` ( 2nd ` y ) ) ) >. )
41	32, 33, 34, 40	fmptco 6040	. . . . . . 7 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( f ` ( g ` ( 2nd ` y ) ) ) >. ) )
42		fvco3 5925	. . . . . . . . . 10 |- ( ( g : (mREx ` T ) --> (mREx ` T ) /\ ( 2nd ` y ) e. (mREx ` T ) ) -> ( ( f o. g ) ` ( 2nd ` y ) ) = ( f ` ( g ` ( 2nd ` y ) ) ) )
43	26, 28, 42	syl2anc 659	. . . . . . . . 9 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( ( f o. g ) ` ( 2nd ` y ) ) = ( f ` ( g ` ( 2nd ` y ) ) ) )
44	43	opeq2d 4210	. . . . . . . 8 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. = <. ( 1st ` y ) , ( f ` ( g ` ( 2nd ` y ) ) ) >. )
45	44	mpteq2dva 4525	. . . . . . 7 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( f ` ( g ` ( 2nd ` y ) ) ) >. ) )
46	41, 45	eqtr4d 2498	. . . . . 6 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) )
47	2	mrsubco 29148	. . . . . . . 8 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( f o. g ) e. ran (mRSubst ` T ) )
48	7	mptex 6118	. . . . . . . 8 |- ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) e. _V
49		eqid 2454	. . . . . . . . 9 |- ( h e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) ) = ( h e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) )
50		fveq1 5847	. . . . . . . . . . 11 |- ( h = ( f o. g ) -> ( h ` ( 2nd ` y ) ) = ( ( f o. g ) ` ( 2nd ` y ) ) )
51	50	opeq2d 4210	. . . . . . . . . 10 |- ( h = ( f o. g ) -> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. = <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. )
52	51	mpteq2dv 4526	. . . . . . . . 9 |- ( h = ( f o. g ) -> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) )
53	49, 52	elrnmpt1s 5239	. . . . . . . 8 |- ( ( ( f o. g ) e. ran (mRSubst ` T ) /\ ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) e. _V ) -> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) e. ran ( h e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) ) )
54	47, 48, 53	sylancl 660	. . . . . . 7 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) e. ran ( h e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) ) )
55	1, 2, 3	elmsubrn 29155	. . . . . . 7 |- ran S = ran ( h e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) )
56	54, 55	syl6eleqr 2553	. . . . . 6 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) e. ran S )
57	46, 56	eqeltrd 2542	. . . . 5 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) e. ran S )
58		coeq1 5149	. . . . . . 7 |- ( F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) -> ( F o. G ) = ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. G ) )
59		coeq2 5150	. . . . . . 7 |- ( G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) -> ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. G ) = ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) )
60	58, 59	sylan9eq 2515	. . . . . 6 |- ( ( F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) -> ( F o. G ) = ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) )
61	60	eleq1d 2523	. . . . 5 |- ( ( F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) -> ( ( F o. G ) e. ran S <-> ( ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) o. ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) e. ran S ) )
62	57, 61	syl5ibrcom 222	. . . 4 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( ( F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) -> ( F o. G ) e. ran S ) )
63	62	rexlimivv 2951	. . 3 |- ( E. f e. ran (mRSubst ` T ) E. g e. ran (mRSubst ` T ) ( F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) -> ( F o. G ) e. ran S )
64	17, 63	sylbir 213	. 2 |- ( ( E. f e. ran (mRSubst ` T ) F = ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) /\ E. g e. ran (mRSubst ` T ) G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) -> ( F o. G ) e. ran S )
65	10, 16, 64	syl2anb 477	1 |- ( ( F e. ran S /\ G e. ran S ) -> ( F o. G ) e. ran S )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   E.wrex 2805   _Vcvv 3106   <.cop 4022   |-> cmpt 4497   X. cxp 4986   ran crn 4989   o. ccom 4992   -->wf 5566   ` cfv 5570   1stc1st 6771   2ndc2nd 6772  mTCcmtc 29091  mRExcmrex 29093  mExcmex 29094  mRSubstcmrsub 29097  mSubstcmsub 29098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-word 12529  df-lsw 12530  df-concat 12531  df-s1 12532  df-substr 12533  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-frmd 16219  df-vrmd 16220  df-mrex 29113  df-mex 29114  df-mrsub 29117  df-msub 29118

This theorem is referenced by:  mclsppslem  29210