Theorem opsrtoslem2 19485

Description: Lemma for opsrtos 19486. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
opsrso.o	|- O = ( ( I ordPwSer R ) ` T )
opsrso.i	|- ( ph -> I e. V )
opsrso.r	|- ( ph -> R e. Toset )
opsrso.t	|- ( ph -> T C_ ( I X. I ) )
opsrso.w	|- ( ph -> T We I )
opsrtoslem.s	|- S = ( I mPwSer R )
opsrtoslem.b	|- B = ( Base ` S )
opsrtoslem.q	|- .< = ( lt ` R )
opsrtoslem.c	|- C = ( T <bag I )
opsrtoslem.d	|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
opsrtoslem.ps	|- ( ps <-> E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) )
opsrtoslem.l	|- .<_ = ( le ` O )

Assertion

Ref	Expression
opsrtoslem2	|- ( ph -> O e. Toset )

Distinct variable groups:   x, y, B   x, w, y, z, C   w, h, x, y, z, I   ph, h, w, x, y, z   w, D, x, y, z   w, .< , x, y, z   w, R, x, y, z   w, T, x, y, z

Allowed substitution hints:   ps( x, y, z, w, h)   B( z, w, h)   C( h)   D( h)   R( h)   S( x, y, z, w, h)   .< ( h)   T( h)   .<_ ( x, y, z, w, h)   O( x, y, z, w, h)   V( x, y, z, w, h)

Proof of Theorem opsrtoslem2

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opsrtoslem.d	. . . . . . . 8 |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }
2		ovex 6678	. . . . . . . 8 |- ( NN0 ^m I ) e. _V
3	1, 2	rabex2 4815	. . . . . . 7 |- D e. _V
4		opsrtoslem.c	. . . . . . . 8 |- C = ( T <bag I )
5		opsrso.i	. . . . . . . 8 |- ( ph -> I e. V )
6		xpexg 6960	. . . . . . . . . 10 |- ( ( I e. V /\ I e. V ) -> ( I X. I ) e. _V )
7	5, 5, 6	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( I X. I ) e. _V )
8		opsrso.t	. . . . . . . . 9 |- ( ph -> T C_ ( I X. I ) )
9	7, 8	ssexd 4805	. . . . . . . 8 |- ( ph -> T e. _V )
10		opsrso.w	. . . . . . . 8 |- ( ph -> T We I )
11	4, 1, 5, 9, 10	ltbwe 19472	. . . . . . 7 |- ( ph -> C We D )
12		opsrso.r	. . . . . . . . 9 |- ( ph -> R e. Toset )
13		eqid 2622	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
14		eqid 2622	. . . . . . . . . . 11 |- ( le ` R ) = ( le ` R )
15		opsrtoslem.q	. . . . . . . . . . 11 |- .< = ( lt ` R )
16	13, 14, 15	tosso 17036	. . . . . . . . . 10 |- ( R e. Toset -> ( R e. Toset <-> ( .< Or ( Base ` R ) /\ ( _I |` ( Base ` R ) ) C_ ( le ` R ) ) ) )
17	16	ibi 256	. . . . . . . . 9 |- ( R e. Toset -> ( .< Or ( Base ` R ) /\ ( _I |` ( Base ` R ) ) C_ ( le ` R ) ) )
18	12, 17	syl 17	. . . . . . . 8 |- ( ph -> ( .< Or ( Base ` R ) /\ ( _I |` ( Base ` R ) ) C_ ( le ` R ) ) )
19	18	simpld 475	. . . . . . 7 |- ( ph -> .< Or ( Base ` R ) )
20		opsrtoslem.ps	. . . . . . . . 9 |- ( ps <-> E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) )
21	20	opabbii 4717	. . . . . . . 8 |- { <. x , y >. | ps } = { <. x , y >. | E. z e. D ( ( x ` z ) .< ( y ` z ) /\ A. w e. D ( w C z -> ( x ` w ) = ( y ` w ) ) ) }
22	21	wemapso 8456	. . . . . . 7 |- ( ( D e. _V /\ C We D /\ .< Or ( Base ` R ) ) -> { <. x , y >. | ps } Or ( ( Base ` R ) ^m D ) )
23	3, 11, 19, 22	mp3an2i 1429	. . . . . 6 |- ( ph -> { <. x , y >. | ps } Or ( ( Base ` R ) ^m D ) )
24		opsrtoslem.s	. . . . . . . 8 |- S = ( I mPwSer R )
25		opsrtoslem.b	. . . . . . . 8 |- B = ( Base ` S )
26	24, 13, 1, 25, 5	psrbas 19378	. . . . . . 7 |- ( ph -> B = ( ( Base ` R ) ^m D ) )
27		soeq2 5055	. . . . . . 7 |- ( B = ( ( Base ` R ) ^m D ) -> ( { <. x , y >. | ps } Or B <-> { <. x , y >. | ps } Or ( ( Base ` R ) ^m D ) ) )
28	26, 27	syl 17	. . . . . 6 |- ( ph -> ( { <. x , y >. | ps } Or B <-> { <. x , y >. | ps } Or ( ( Base ` R ) ^m D ) ) )
29	23, 28	mpbird 247	. . . . 5 |- ( ph -> { <. x , y >. | ps } Or B )
30		soinxp 5183	. . . . 5 |- ( { <. x , y >. | ps } Or B <-> ( { <. x , y >. | ps } i^i ( B X. B ) ) Or B )
31	29, 30	sylib 208	. . . 4 |- ( ph -> ( { <. x , y >. | ps } i^i ( B X. B ) ) Or B )
32		opsrso.o	. . . . . . . 8 |- O = ( ( I ordPwSer R ) ` T )
33		fvex 6201	. . . . . . . 8 |- ( ( I ordPwSer R ) ` T ) e. _V
34	32, 33	eqeltri 2697	. . . . . . 7 |- O e. _V
35		opsrtoslem.l	. . . . . . . 8 |- .<_ = ( le ` O )
36		eqid 2622	. . . . . . . 8 |- ( lt ` O ) = ( lt ` O )
37	35, 36	pltfval 16959	. . . . . . 7 |- ( O e. _V -> ( lt ` O ) = ( .<_ \ _I ) )
38	34, 37	ax-mp 5	. . . . . 6 |- ( lt ` O ) = ( .<_ \ _I )
39		difundir 3880	. . . . . . . 8 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) u. ( _I |` B ) ) \ _I ) = ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I ) u. ( ( _I |` B ) \ _I ) )
40		resss 5422	. . . . . . . . . 10 |- ( _I |` B ) C_ _I
41		ssdif0 3942	. . . . . . . . . 10 |- ( ( _I |` B ) C_ _I <-> ( ( _I |` B ) \ _I ) = (/) )
42	40, 41	mpbi 220	. . . . . . . . 9 |- ( ( _I |` B ) \ _I ) = (/)
43	42	uneq2i 3764	. . . . . . . 8 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I ) u. ( ( _I |` B ) \ _I ) ) = ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I ) u. (/) )
44		un0 3967	. . . . . . . 8 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I ) u. (/) ) = ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I )
45	39, 43, 44	3eqtri 2648	. . . . . . 7 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) u. ( _I |` B ) ) \ _I ) = ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I )
46	32, 5, 12, 8, 10, 24, 25, 15, 4, 1, 20, 35	opsrtoslem1 19484	. . . . . . . 8 |- ( ph -> .<_ = ( ( { <. x , y >. | ps } i^i ( B X. B ) ) u. ( _I |` B ) ) )
47	46	difeq1d 3727	. . . . . . 7 |- ( ph -> ( .<_ \ _I ) = ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) u. ( _I |` B ) ) \ _I ) )
48		inss2 3834	. . . . . . . . . . . 12 |- ( { <. x , y >. | ps } i^i ( B X. B ) ) C_ ( B X. B )
49		relxp 5227	. . . . . . . . . . . 12 |- Rel ( B X. B )
50		relss 5206	. . . . . . . . . . . 12 |- ( ( { <. x , y >. | ps } i^i ( B X. B ) ) C_ ( B X. B ) -> ( Rel ( B X. B ) -> Rel ( { <. x , y >. | ps } i^i ( B X. B ) ) ) )
51	48, 49, 50	mp2 9	. . . . . . . . . . 11 |- Rel ( { <. x , y >. | ps } i^i ( B X. B ) )
52	51	a1i 11	. . . . . . . . . 10 |- ( ph -> Rel ( { <. x , y >. | ps } i^i ( B X. B ) ) )
53		df-br 4654	. . . . . . . . . . . . . 14 |- ( a _I b <-> <. a , b >. e. _I )
54		vex 3203	. . . . . . . . . . . . . . 15 |- b e. _V
55	54	ideq 5274	. . . . . . . . . . . . . 14 |- ( a _I b <-> a = b )
56	53, 55	bitr3i 266	. . . . . . . . . . . . 13 |- ( <. a , b >. e. _I <-> a = b )
57		brin 4704	. . . . . . . . . . . . . . . . . 18 |- ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) a <-> ( a { <. x , y >. | ps } a /\ a ( B X. B ) a ) )
58	57	simprbi 480	. . . . . . . . . . . . . . . . 17 |- ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) a -> a ( B X. B ) a )
59		brxp 5147	. . . . . . . . . . . . . . . . . 18 |- ( a ( B X. B ) a <-> ( a e. B /\ a e. B ) )
60	59	simprbi 480	. . . . . . . . . . . . . . . . 17 |- ( a ( B X. B ) a -> a e. B )
61	58, 60	syl 17	. . . . . . . . . . . . . . . 16 |- ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) a -> a e. B )
62		sonr 5056	. . . . . . . . . . . . . . . . 17 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) Or B /\ a e. B ) -> -. a ( { <. x , y >. | ps } i^i ( B X. B ) ) a )
63	62	ex 450	. . . . . . . . . . . . . . . 16 |- ( ( { <. x , y >. | ps } i^i ( B X. B ) ) Or B -> ( a e. B -> -. a ( { <. x , y >. | ps } i^i ( B X. B ) ) a ) )
64	31, 61, 63	syl2im 40	. . . . . . . . . . . . . . 15 |- ( ph -> ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) a -> -. a ( { <. x , y >. | ps } i^i ( B X. B ) ) a ) )
65	64	pm2.01d 181	. . . . . . . . . . . . . 14 |- ( ph -> -. a ( { <. x , y >. | ps } i^i ( B X. B ) ) a )
66		breq2 4657	. . . . . . . . . . . . . . . 16 |- ( a = b -> ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) a <-> a ( { <. x , y >. | ps } i^i ( B X. B ) ) b ) )
67		df-br 4654	. . . . . . . . . . . . . . . 16 |- ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) b <-> <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) )
68	66, 67	syl6bb 276	. . . . . . . . . . . . . . 15 |- ( a = b -> ( a ( { <. x , y >. | ps } i^i ( B X. B ) ) a <-> <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) ) )
69	68	notbid 308	. . . . . . . . . . . . . 14 |- ( a = b -> ( -. a ( { <. x , y >. | ps } i^i ( B X. B ) ) a <-> -. <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) ) )
70	65, 69	syl5ibcom 235	. . . . . . . . . . . . 13 |- ( ph -> ( a = b -> -. <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) ) )
71	56, 70	syl5bi 232	. . . . . . . . . . . 12 |- ( ph -> ( <. a , b >. e. _I -> -. <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) ) )
72	71	con2d 129	. . . . . . . . . . 11 |- ( ph -> ( <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) -> -. <. a , b >. e. _I ) )
73		opex 4932	. . . . . . . . . . . 12 |- <. a , b >. e. _V
74		eldif 3584	. . . . . . . . . . . 12 |- ( <. a , b >. e. ( _V \ _I ) <-> ( <. a , b >. e. _V /\ -. <. a , b >. e. _I ) )
75	73, 74	mpbiran 953	. . . . . . . . . . 11 |- ( <. a , b >. e. ( _V \ _I ) <-> -. <. a , b >. e. _I )
76	72, 75	syl6ibr 242	. . . . . . . . . 10 |- ( ph -> ( <. a , b >. e. ( { <. x , y >. | ps } i^i ( B X. B ) ) -> <. a , b >. e. ( _V \ _I ) ) )
77	52, 76	relssdv 5212	. . . . . . . . 9 |- ( ph -> ( { <. x , y >. | ps } i^i ( B X. B ) ) C_ ( _V \ _I ) )
78		disj2 4024	. . . . . . . . 9 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) i^i _I ) = (/) <-> ( { <. x , y >. | ps } i^i ( B X. B ) ) C_ ( _V \ _I ) )
79	77, 78	sylibr 224	. . . . . . . 8 |- ( ph -> ( ( { <. x , y >. | ps } i^i ( B X. B ) ) i^i _I ) = (/) )
80		disj3 4021	. . . . . . . 8 |- ( ( ( { <. x , y >. | ps } i^i ( B X. B ) ) i^i _I ) = (/) <-> ( { <. x , y >. | ps } i^i ( B X. B ) ) = ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I ) )
81	79, 80	sylib 208	. . . . . . 7 |- ( ph -> ( { <. x , y >. | ps } i^i ( B X. B ) ) = ( ( { <. x , y >. | ps } i^i ( B X. B ) ) \ _I ) )
82	45, 47, 81	3eqtr4a 2682	. . . . . 6 |- ( ph -> ( .<_ \ _I ) = ( { <. x , y >. | ps } i^i ( B X. B ) ) )
83	38, 82	syl5eq 2668	. . . . 5 |- ( ph -> ( lt ` O ) = ( { <. x , y >. | ps } i^i ( B X. B ) ) )
84		soeq1 5054	. . . . 5 |- ( ( lt ` O ) = ( { <. x , y >. | ps } i^i ( B X. B ) ) -> ( ( lt ` O ) Or B <-> ( { <. x , y >. | ps } i^i ( B X. B ) ) Or B ) )
85	83, 84	syl 17	. . . 4 |- ( ph -> ( ( lt ` O ) Or B <-> ( { <. x , y >. | ps } i^i ( B X. B ) ) Or B ) )
86	31, 85	mpbird 247	. . 3 |- ( ph -> ( lt ` O ) Or B )
87	24, 32, 8	opsrbas 19479	. . . . 5 |- ( ph -> ( Base ` S ) = ( Base ` O ) )
88	25, 87	syl5eq 2668	. . . 4 |- ( ph -> B = ( Base ` O ) )
89		soeq2 5055	. . . 4 |- ( B = ( Base ` O ) -> ( ( lt ` O ) Or B <-> ( lt ` O ) Or ( Base ` O ) ) )
90	88, 89	syl 17	. . 3 |- ( ph -> ( ( lt ` O ) Or B <-> ( lt ` O ) Or ( Base ` O ) ) )
91	86, 90	mpbid 222	. 2 |- ( ph -> ( lt ` O ) Or ( Base ` O ) )
92	88	reseq2d 5396	. . . 4 |- ( ph -> ( _I |` B ) = ( _I |` ( Base ` O ) ) )
93		ssun2 3777	. . . 4 |- ( _I |` B ) C_ ( ( { <. x , y >. | ps } i^i ( B X. B ) ) u. ( _I |` B ) )
94	92, 93	syl6eqssr 3656	. . 3 |- ( ph -> ( _I |` ( Base ` O ) ) C_ ( ( { <. x , y >. | ps } i^i ( B X. B ) ) u. ( _I |` B ) ) )
95	94, 46	sseqtr4d 3642	. 2 |- ( ph -> ( _I |` ( Base ` O ) ) C_ .<_ )
96		eqid 2622	. . . 4 |- ( Base ` O ) = ( Base ` O )
97	96, 35, 36	tosso 17036	. . 3 |- ( O e. _V -> ( O e. Toset <-> ( ( lt ` O ) Or ( Base ` O ) /\ ( _I |` ( Base ` O ) ) C_ .<_ ) ) )
98	34, 97	ax-mp 5	. 2 |- ( O e. Toset <-> ( ( lt ` O ) Or ( Base ` O ) /\ ( _I |` ( Base ` O ) ) C_ .<_ ) )
99	91, 95, 98	sylanbrc 698	1 |- ( ph -> O e. Toset )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   {copab 4712   _I cid 5023   Or wor 5034   We wwe 5072   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   NNcn 11020   NN0cn0 11292   Basecbs 15857   lecple 15948   ltcplt 16941  Tosetctos 17033   mPwSer cmps 19351   <_bag cltb 19354   ordPwSer copws 19355

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-inf2 8538  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-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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-ple 15961  df-preset 16928  df-poset 16946  df-plt 16958  df-toset 17034  df-psr 19356  df-ltbag 19359  df-opsr 19360

This theorem is referenced by:  opsrtos  19486