Theorem mclspps 30222

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 30207.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mclspps.d	|- D = (mDV ` T )
mclspps.e	|- E = (mEx ` T )
mclspps.c	|- C = (mCls ` T )
mclspps.1	|- ( ph -> T e. mFS )
mclspps.2	|- ( ph -> K C_ D )
mclspps.3	|- ( ph -> B C_ E )
mclspps.j	|- J = (mPPSt ` T )
mclspps.l	|- L = (mSubst ` T )
mclspps.v	|- V = (mVR ` T )
mclspps.h	|- H = (mVH ` T )
mclspps.w	|- W = (mVars ` T )
mclspps.4	|- ( ph -> <. M , O , P >. e. J )
mclspps.5	|- ( ph -> S e. ran L )
mclspps.6	|- ( ( ph /\ x e. O ) -> ( S ` x ) e. ( K C B ) )
mclspps.7	|- ( ( ph /\ v e. V ) -> ( S ` ( H ` v ) ) e. ( K C B ) )
mclspps.8	|- ( ( ph /\ ( x M y /\ a e. ( W ` ( S ` ( H ` x ) ) ) /\ b e. ( W ` ( S ` ( H ` y ) ) ) ) ) -> a K b )

Assertion

Ref	Expression
mclspps	|- ( ph -> ( S ` P ) e. ( K C B ) )

Distinct variable groups:   v, E   a, b, v, x, y, H   v, V   K, a, b, v, x, y   T, a, b, v, x, y   L, a, b, v, x, y   S, a, b, v, x, y   B, a, b, v, x, y   W, a, b, v, x, y   C, a, b, v, x, y   M, a, b, v, x, y   v, O, x   ph, a, b, v, x, y

Allowed substitution hints:   D( x, y, v, a, b)   P( x, y, v, a, b)   E( x, y, a, b)   J( x, y, v, a, b)   O( y, a, b)   V( x, y, a, b)

Proof of Theorem mclspps

Dummy variables m o p s w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mclspps.5	. . . 4 |- ( ph -> S e. ran L )
2		mclspps.l	. . . . 5 |- L = (mSubst ` T )
3		mclspps.e	. . . . 5 |- E = (mEx ` T )
4	2, 3	msubf 30170	. . . 4 |- ( S e. ran L -> S : E --> E )
5	1, 4	syl 17	. . 3 |- ( ph -> S : E --> E )
6		ffn 5728	. . 3 |- ( S : E --> E -> S Fn E )
7	5, 6	syl 17	. 2 |- ( ph -> S Fn E )
8		mclspps.d	. . . 4 |- D = (mDV ` T )
9		mclspps.c	. . . 4 |- C = (mCls ` T )
10		mclspps.1	. . . 4 |- ( ph -> T e. mFS )
11		eqid 2451	. . . . . . . . 9 |- (mPreSt ` T ) = (mPreSt ` T )
12		mclspps.j	. . . . . . . . 9 |- J = (mPPSt ` T )
13	11, 12	mppspst 30212	. . . . . . . 8 |- J C_ (mPreSt ` T )
14		mclspps.4	. . . . . . . 8 |- ( ph -> <. M , O , P >. e. J )
15	13, 14	sseldi 3430	. . . . . . 7 |- ( ph -> <. M , O , P >. e. (mPreSt ` T ) )
16	8, 3, 11	elmpst 30174	. . . . . . 7 |- ( <. M , O , P >. e. (mPreSt ` T ) <-> ( ( M C_ D /\ `' M = M ) /\ ( O C_ E /\ O e. Fin ) /\ P e. E ) )
17	15, 16	sylib 200	. . . . . 6 |- ( ph -> ( ( M C_ D /\ `' M = M ) /\ ( O C_ E /\ O e. Fin ) /\ P e. E ) )
18	17	simp1d 1020	. . . . 5 |- ( ph -> ( M C_ D /\ `' M = M ) )
19	18	simpld 461	. . . 4 |- ( ph -> M C_ D )
20	17	simp2d 1021	. . . . 5 |- ( ph -> ( O C_ E /\ O e. Fin ) )
21	20	simpld 461	. . . 4 |- ( ph -> O C_ E )
22		eqid 2451	. . . 4 |- (mAx ` T ) = (mAx ` T )
23		mclspps.v	. . . 4 |- V = (mVR ` T )
24		mclspps.h	. . . 4 |- H = (mVH ` T )
25		mclspps.w	. . . 4 |- W = (mVars ` T )
26		mclspps.6	. . . . . 6 |- ( ( ph /\ x e. O ) -> ( S ` x ) e. ( K C B ) )
27	26	ralrimiva 2802	. . . . 5 |- ( ph -> A. x e. O ( S ` x ) e. ( K C B ) )
28		ffun 5731	. . . . . . 7 |- ( S : E --> E -> Fun S )
29	5, 28	syl 17	. . . . . 6 |- ( ph -> Fun S )
30		fdm 5733	. . . . . . . 8 |- ( S : E --> E -> dom S = E )
31	5, 30	syl 17	. . . . . . 7 |- ( ph -> dom S = E )
32	21, 31	sseqtr4d 3469	. . . . . 6 |- ( ph -> O C_ dom S )
33		funimass5 5999	. . . . . 6 |- ( ( Fun S /\ O C_ dom S ) -> ( O C_ ( `' S " ( K C B ) ) <-> A. x e. O ( S ` x ) e. ( K C B ) ) )
34	29, 32, 33	syl2anc 667	. . . . 5 |- ( ph -> ( O C_ ( `' S " ( K C B ) ) <-> A. x e. O ( S ` x ) e. ( K C B ) ) )
35	27, 34	mpbird 236	. . . 4 |- ( ph -> O C_ ( `' S " ( K C B ) ) )
36	23, 3, 24	mvhf 30196	. . . . . . 7 |- ( T e. mFS -> H : V --> E )
37	10, 36	syl 17	. . . . . 6 |- ( ph -> H : V --> E )
38	37	ffvelrnda 6022	. . . . 5 |- ( ( ph /\ v e. V ) -> ( H ` v ) e. E )
39		mclspps.7	. . . . 5 |- ( ( ph /\ v e. V ) -> ( S ` ( H ` v ) ) e. ( K C B ) )
40		elpreima 6002	. . . . . . 7 |- ( S Fn E -> ( ( H ` v ) e. ( `' S " ( K C B ) ) <-> ( ( H ` v ) e. E /\ ( S ` ( H ` v ) ) e. ( K C B ) ) ) )
41	7, 40	syl 17	. . . . . 6 |- ( ph -> ( ( H ` v ) e. ( `' S " ( K C B ) ) <-> ( ( H ` v ) e. E /\ ( S ` ( H ` v ) ) e. ( K C B ) ) ) )
42	41	adantr 467	. . . . 5 |- ( ( ph /\ v e. V ) -> ( ( H ` v ) e. ( `' S " ( K C B ) ) <-> ( ( H ` v ) e. E /\ ( S ` ( H ` v ) ) e. ( K C B ) ) ) )
43	38, 39, 42	mpbir2and 933	. . . 4 |- ( ( ph /\ v e. V ) -> ( H ` v ) e. ( `' S " ( K C B ) ) )
44	10	3ad2ant1 1029	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> T e. mFS )
45		mclspps.2	. . . . . 6 |- ( ph -> K C_ D )
46	45	3ad2ant1 1029	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> K C_ D )
47		mclspps.3	. . . . . 6 |- ( ph -> B C_ E )
48	47	3ad2ant1 1029	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> B C_ E )
49	14	3ad2ant1 1029	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> <. M , O , P >. e. J )
50	1	3ad2ant1 1029	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> S e. ran L )
51	26	3ad2antl1 1170	. . . . 5 |- ( ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) /\ x e. O ) -> ( S ` x ) e. ( K C B ) )
52	39	3ad2antl1 1170	. . . . 5 |- ( ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) /\ v e. V ) -> ( S ` ( H ` v ) ) e. ( K C B ) )
53		mclspps.8	. . . . . 6 |- ( ( ph /\ ( x M y /\ a e. ( W ` ( S ` ( H ` x ) ) ) /\ b e. ( W ` ( S ` ( H ` y ) ) ) ) ) -> a K b )
54	53	3ad2antl1 1170	. . . . 5 |- ( ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) /\ ( x M y /\ a e. ( W ` ( S ` ( H ` x ) ) ) /\ b e. ( W ` ( S ` ( H ` y ) ) ) ) ) -> a K b )
55		simp21 1041	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> <. m , o , p >. e. (mAx ` T ) )
56		simp22 1042	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> s e. ran L )
57		simp23 1043	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) )
58		simp3 1010	. . . . 5 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) )
59	8, 3, 9, 44, 46, 48, 12, 2, 23, 24, 25, 49, 50, 51, 52, 54, 55, 56, 57, 58	mclsppslem 30221	. . . 4 |- ( ( ph /\ ( <. m , o , p >. e. (mAx ` T ) /\ s e. ran L /\ ( s " ( o u. ran H ) ) C_ ( `' S " ( K C B ) ) ) /\ A. z A. w ( z m w -> ( ( W ` ( s ` ( H ` z ) ) ) X. ( W ` ( s ` ( H ` w ) ) ) ) C_ M ) ) -> ( s ` p ) e. ( `' S " ( K C B ) ) )
60	8, 3, 9, 10, 19, 21, 22, 2, 23, 24, 25, 35, 43, 59	mclsind 30208	. . 3 |- ( ph -> ( M C O ) C_ ( `' S " ( K C B ) ) )
61	11, 12, 9	elmpps 30211	. . . . 5 |- ( <. M , O , P >. e. J <-> ( <. M , O , P >. e. (mPreSt ` T ) /\ P e. ( M C O ) ) )
62	61	simprbi 466	. . . 4 |- ( <. M , O , P >. e. J -> P e. ( M C O ) )
63	14, 62	syl 17	. . 3 |- ( ph -> P e. ( M C O ) )
64	60, 63	sseldd 3433	. 2 |- ( ph -> P e. ( `' S " ( K C B ) ) )
65		elpreima 6002	. . 3 |- ( S Fn E -> ( P e. ( `' S " ( K C B ) ) <-> ( P e. E /\ ( S ` P ) e. ( K C B ) ) ) )
66	65	simplbda 630	. 2 |- ( ( S Fn E /\ P e. ( `' S " ( K C B ) ) ) -> ( S ` P ) e. ( K C B ) )
67	7, 64, 66	syl2anc 667	1 |- ( ph -> ( S ` P ) e. ( K C B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   A.wal 1442   = wceq 1444   e. wcel 1887   A.wral 2737   u. cun 3402   C_ wss 3404   <.cotp 3976   class class class wbr 4402   X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Fun wfun 5576   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569  mVRcmvar 30099  mAxcmax 30103  mExcmex 30105  mDVcmdv 30106  mVarscmvrs 30107  mSubstcmsub 30109  mVHcmvh 30110  mPreStcmpst 30111  mFScmfs 30114  mClscmcls 30115  mPPStcmpps 30116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-frmd 16633  df-vrmd 16634  df-mrex 30124  df-mex 30125  df-mdv 30126  df-mvrs 30127  df-mrsub 30128  df-msub 30129  df-mvh 30130  df-mpst 30131  df-msr 30132  df-msta 30133  df-mfs 30134  df-mcls 30135  df-mpps 30136

This theorem is referenced by: (None)