Theorem msubco 31428

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 2622	. . . . 5 |- (mEx ` T ) = (mEx ` T )
2		eqid 2622	. . . . 5 |- (mRSubst ` T ) = (mRSubst ` T )
3		msubco.s	. . . . 5 |- S = (mSubst ` T )
4	1, 2, 3	elmsubrn 31425	. . . 4 |- ran S = ran ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) )
5	4	eleq2i 2693	. . 3 |- ( F e. ran S <-> F e. ran ( f e. ran (mRSubst ` T ) |-> ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) ) )
6		eqid 2622	. . . 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 6201	. . . . 5 |- (mEx ` T ) e. _V
8	7	mptex 6486	. . . 4 |- ( x e. (mEx ` T ) |-> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. ) e. _V
9	6, 8	elrnmpti 5376	. . 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 264	. 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 31425	. . . 4 |- ran S = ran ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
12	11	eleq2i 2693	. . 3 |- ( G e. ran S <-> G e. ran ( g e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) ) )
13		eqid 2622	. . . 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 6486	. . . 4 |- ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) e. _V
15	13, 14	elrnmpti 5376	. . 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 264	. 2 |- ( G e. ran S <-> E. g e. ran (mRSubst ` T ) G = ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. ) )
17		reeanv 3107	. . 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 477	. . . . . . . . . . . 12 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> y e. (mEx ` T ) )
19		eqid 2622	. . . . . . . . . . . . 13 |- (mTC ` T ) = (mTC ` T )
20		eqid 2622	. . . . . . . . . . . . 13 |- (mREx ` T ) = (mREx ` T )
21	19, 1, 20	mexval 31399	. . . . . . . . . . . 12 |- (mEx ` T ) = ( (mTC ` T ) X. (mREx ` T ) )
22	18, 21	syl6eleq 2711	. . . . . . . . . . 11 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> y e. ( (mTC ` T ) X. (mREx ` T ) ) )
23		xp1st 7198	. . . . . . . . . . 11 |- ( y e. ( (mTC ` T ) X. (mREx ` T ) ) -> ( 1st ` y ) e. (mTC ` T ) )
24	22, 23	syl 17	. . . . . . . . . 10 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( 1st ` y ) e. (mTC ` T ) )
25	2, 20	mrsubf 31414	. . . . . . . . . . . 12 |- ( g e. ran (mRSubst ` T ) -> g : (mREx ` T ) --> (mREx ` T ) )
26	25	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> g : (mREx ` T ) --> (mREx ` T ) )
27		xp2nd 7199	. . . . . . . . . . . 12 |- ( y e. ( (mTC ` T ) X. (mREx ` T ) ) -> ( 2nd ` y ) e. (mREx ` T ) )
28	22, 27	syl 17	. . . . . . . . . . 11 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( 2nd ` y ) e. (mREx ` T ) )
29	26, 28	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) /\ y e. (mEx ` T ) ) -> ( g ` ( 2nd ` y ) ) e. (mREx ` T ) )
30		opelxpi 5148	. . . . . . . . . 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 693	. . . . . . . . 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 2712	. . . . . . . 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 2623	. . . . . . . 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 2623	. . . . . . . 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 6201	. . . . . . . . . 10 |- ( 1st ` y ) e. _V
36		fvex 6201	. . . . . . . . . 10 |- ( g ` ( 2nd ` y ) ) e. _V
37	35, 36	op1std 7178	. . . . . . . . 9 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> ( 1st ` x ) = ( 1st ` y ) )
38	35, 36	op2ndd 7179	. . . . . . . . . 10 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> ( 2nd ` x ) = ( g ` ( 2nd ` y ) ) )
39	38	fveq2d 6195	. . . . . . . . 9 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> ( f ` ( 2nd ` x ) ) = ( f ` ( g ` ( 2nd ` y ) ) ) )
40	37, 39	opeq12d 4410	. . . . . . . 8 |- ( x = <. ( 1st ` y ) , ( g ` ( 2nd ` y ) ) >. -> <. ( 1st ` x ) , ( f ` ( 2nd ` x ) ) >. = <. ( 1st ` y ) , ( f ` ( g ` ( 2nd ` y ) ) ) >. )
41	32, 33, 34, 40	fmptco 6396	. . . . . . 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 6275	. . . . . . . . . 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 693	. . . . . . . . 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 4409	. . . . . . . 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 4744	. . . . . . 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 2659	. . . . . 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 31418	. . . . . . . 8 |- ( ( f e. ran (mRSubst ` T ) /\ g e. ran (mRSubst ` T ) ) -> ( f o. g ) e. ran (mRSubst ` T ) )
48	7	mptex 6486	. . . . . . . 8 |- ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. ) e. _V
49		eqid 2622	. . . . . . . . 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 6190	. . . . . . . . . . 11 |- ( h = ( f o. g ) -> ( h ` ( 2nd ` y ) ) = ( ( f o. g ) ` ( 2nd ` y ) ) )
51	50	opeq2d 4409	. . . . . . . . . 10 |- ( h = ( f o. g ) -> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. = <. ( 1st ` y ) , ( ( f o. g ) ` ( 2nd ` y ) ) >. )
52	51	mpteq2dv 4745	. . . . . . . . 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 5373	. . . . . . . 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 694	. . . . . . 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 31425	. . . . . . 7 |- ran S = ran ( h e. ran (mRSubst ` T ) |-> ( y e. (mEx ` T ) |-> <. ( 1st ` y ) , ( h ` ( 2nd ` y ) ) >. ) )
56	54, 55	syl6eleqr 2712	. . . . . 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 2701	. . . . 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 5279	. . . . . . 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 5280	. . . . . . 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 2676	. . . . . 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 2686	. . . . 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 237	. . . 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 3036	. . 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 225	. 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 496	1 |- ( ( F e. ran S /\ G e. ran S ) -> ( F o. G ) e. ran S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  mTCcmtc 31361  mRExcmrex 31363  mExcmex 31364  mRSubstcmrsub 31367  mSubstcmsub 31368

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

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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-frmd 17386  df-vrmd 17387  df-mrex 31383  df-mex 31384  df-mrsub 31387  df-msub 31388

This theorem is referenced by:  mclsppslem  31480