Theorem mclspps 31481

Description: The closure is closed under application of provable pre-statements. (Compare mclsax 31466.) 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 31429	. . . 4 |- ( S e. ran L -> S : E --> E )
5	1, 4	syl 17	. . 3 |- ( ph -> S : E --> E )
6		ffn 6045	. . 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 2622	. . . . . . . . 9 |- (mPreSt ` T ) = (mPreSt ` T )
12		mclspps.j	. . . . . . . . 9 |- J = (mPPSt ` T )
13	11, 12	mppspst 31471	. . . . . . . 8 |- J C_ (mPreSt ` T )
14		mclspps.4	. . . . . . . 8 |- ( ph -> <. M , O , P >. e. J )
15	13, 14	sseldi 3601	. . . . . . 7 |- ( ph -> <. M , O , P >. e. (mPreSt ` T ) )
16	8, 3, 11	elmpst 31433	. . . . . . 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 208	. . . . . 6 |- ( ph -> ( ( M C_ D /\ `' M = M ) /\ ( O C_ E /\ O e. Fin ) /\ P e. E ) )
18	17	simp1d 1073	. . . . 5 |- ( ph -> ( M C_ D /\ `' M = M ) )
19	18	simpld 475	. . . 4 |- ( ph -> M C_ D )
20	17	simp2d 1074	. . . . 5 |- ( ph -> ( O C_ E /\ O e. Fin ) )
21	20	simpld 475	. . . 4 |- ( ph -> O C_ E )
22		eqid 2622	. . . 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 2966	. . . . 5 |- ( ph -> A. x e. O ( S ` x ) e. ( K C B ) )
28		ffun 6048	. . . . . . 7 |- ( S : E --> E -> Fun S )
29	5, 28	syl 17	. . . . . 6 |- ( ph -> Fun S )
30		fdm 6051	. . . . . . . 8 |- ( S : E --> E -> dom S = E )
31	5, 30	syl 17	. . . . . . 7 |- ( ph -> dom S = E )
32	21, 31	sseqtr4d 3642	. . . . . 6 |- ( ph -> O C_ dom S )
33		funimass5 6334	. . . . . 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 693	. . . . 5 |- ( ph -> ( O C_ ( `' S " ( K C B ) ) <-> A. x e. O ( S ` x ) e. ( K C B ) ) )
35	27, 34	mpbird 247	. . . 4 |- ( ph -> O C_ ( `' S " ( K C B ) ) )
36	23, 3, 24	mvhf 31455	. . . . . . 7 |- ( T e. mFS -> H : V --> E )
37	10, 36	syl 17	. . . . . 6 |- ( ph -> H : V --> E )
38	37	ffvelrnda 6359	. . . . 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 6337	. . . . . . 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 481	. . . . 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 957	. . . 4 |- ( ( ph /\ v e. V ) -> ( H ` v ) e. ( `' S " ( K C B ) ) )
44	10	3ad2ant1 1082	. . . . 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 1082	. . . . 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 1082	. . . . 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 1082	. . . . 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 1082	. . . . 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 1223	. . . . 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 1223	. . . . 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 1223	. . . . 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 1094	. . . . 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 1095	. . . . 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 1096	. . . . 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 1063	. . . . 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 31480	. . . 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 31467	. . 3 |- ( ph -> ( M C O ) C_ ( `' S " ( K C B ) ) )
61	11, 12, 9	elmpps 31470	. . . . 5 |- ( <. M , O , P >. e. J <-> ( <. M , O , P >. e. (mPreSt ` T ) /\ P e. ( M C O ) ) )
62	61	simprbi 480	. . . 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 3604	. 2 |- ( ph -> P e. ( `' S " ( K C B ) ) )
65		elpreima 6337	. . 3 |- ( S Fn E -> ( P e. ( `' S " ( K C B ) ) <-> ( P e. E /\ ( S ` P ) e. ( K C B ) ) ) )
66	65	simplbda 654	. 2 |- ( ( S Fn E /\ P e. ( `' S " ( K C B ) ) ) -> ( S ` P ) e. ( K C B ) )
67	7, 64, 66	syl2anc 693	1 |- ( ph -> ( S ` P ) e. ( K C B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   C_ wss 3574   <.cotp 4185   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955  mVRcmvar 31358  mAxcmax 31362  mExcmex 31364  mDVcmdv 31365  mVarscmvrs 31366  mSubstcmsub 31368  mVHcmvh 31369  mPreStcmpst 31370  mFScmfs 31373  mClscmcls 31374  mPPStcmpps 31375

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-fal 1489  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-ot 4186  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-mdv 31385  df-mvrs 31386  df-mrsub 31387  df-msub 31388  df-mvh 31389  df-mpst 31390  df-msr 31391  df-msta 31392  df-mfs 31393  df-mcls 31394  df-mpps 31395

This theorem is referenced by: (None)