Theorem om1val 22830

Description: The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)

Hypotheses

Ref	Expression
om1val.o	|- O = ( J Om1 Y )
om1val.b	|- ( ph -> B = { f e. ( II Cn J ) | ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) } )
om1val.p	|- ( ph -> .+ = ( *p ` J ) )
om1val.k	|- ( ph -> K = ( J ^ko II ) )
om1val.j	|- ( ph -> J e. (TopOn ` X ) )
om1val.y	|- ( ph -> Y e. X )

Assertion

Ref	Expression
om1val	|- ( ph -> O = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , K >. } )

Distinct variable groups:   f, J   ph, f   f, Y

Allowed substitution hints:   B( f)   .+ ( f)   K( f)   O( f)   X( f)

Proof of Theorem om1val

Dummy variables y j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		om1val.o	. 2 |- O = ( J Om1 Y )
2		df-om1 22806	. . . 4 |- Om1 = ( j e. Top , y e. U. j |-> { <. ( Base ` ndx ) , { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } >. , <. ( +g ` ndx ) , ( *p ` j ) >. , <. (TopSet ` ndx ) , ( j ^ko II ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> Om1 = ( j e. Top , y e. U. j |-> { <. ( Base ` ndx ) , { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } >. , <. ( +g ` ndx ) , ( *p ` j ) >. , <. (TopSet ` ndx ) , ( j ^ko II ) >. } ) )
4		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> j = J )
5	4	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( II Cn j ) = ( II Cn J ) )
6		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> y = Y )
7	6	eqeq2d 2632	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( ( f ` 0 ) = y <-> ( f ` 0 ) = Y ) )
8	6	eqeq2d 2632	. . . . . . . 8 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( ( f ` 1 ) = y <-> ( f ` 1 ) = Y ) )
9	7, 8	anbi12d 747	. . . . . . 7 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) <-> ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) ) )
10	5, 9	rabeqbidv 3195	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } = { f e. ( II Cn J ) | ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) } )
11		om1val.b	. . . . . . 7 |- ( ph -> B = { f e. ( II Cn J ) | ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) } )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> B = { f e. ( II Cn J ) | ( ( f ` 0 ) = Y /\ ( f ` 1 ) = Y ) } )
13	10, 12	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } = B )
14	13	opeq2d 4409	. . . 4 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> <. ( Base ` ndx ) , { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } >. = <. ( Base ` ndx ) , B >. )
15	4	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( *p ` j ) = ( *p ` J ) )
16		om1val.p	. . . . . . 7 |- ( ph -> .+ = ( *p ` J ) )
17	16	adantr 481	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> .+ = ( *p ` J ) )
18	15, 17	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( *p ` j ) = .+ )
19	18	opeq2d 4409	. . . 4 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> <. ( +g ` ndx ) , ( *p ` j ) >. = <. ( +g ` ndx ) , .+ >. )
20	4	oveq1d 6665	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( j ^ko II ) = ( J ^ko II ) )
21		om1val.k	. . . . . . 7 |- ( ph -> K = ( J ^ko II ) )
22	21	adantr 481	. . . . . 6 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> K = ( J ^ko II ) )
23	20, 22	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> ( j ^ko II ) = K )
24	23	opeq2d 4409	. . . 4 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> <. (TopSet ` ndx ) , ( j ^ko II ) >. = <. (TopSet ` ndx ) , K >. )
25	14, 19, 24	tpeq123d 4283	. . 3 |- ( ( ph /\ ( j = J /\ y = Y ) ) -> { <. ( Base ` ndx ) , { f e. ( II Cn j ) | ( ( f ` 0 ) = y /\ ( f ` 1 ) = y ) } >. , <. ( +g ` ndx ) , ( *p ` j ) >. , <. (TopSet ` ndx ) , ( j ^ko II ) >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , K >. } )
26		unieq 4444	. . . . 5 |- ( j = J -> U. j = U. J )
27	26	adantl 482	. . . 4 |- ( ( ph /\ j = J ) -> U. j = U. J )
28		om1val.j	. . . . . 6 |- ( ph -> J e. (TopOn ` X ) )
29		toponuni 20719	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
30	28, 29	syl 17	. . . . 5 |- ( ph -> X = U. J )
31	30	adantr 481	. . . 4 |- ( ( ph /\ j = J ) -> X = U. J )
32	27, 31	eqtr4d 2659	. . 3 |- ( ( ph /\ j = J ) -> U. j = X )
33		topontop 20718	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
34	28, 33	syl 17	. . 3 |- ( ph -> J e. Top )
35		om1val.y	. . 3 |- ( ph -> Y e. X )
36		tpex 6957	. . . 4 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , K >. } e. _V
37	36	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , K >. } e. _V )
38	3, 25, 32, 34, 35, 37	ovmpt2dx 6787	. 2 |- ( ph -> ( J Om1 Y ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , K >. } )
39	1, 38	syl5eq 2668	1 |- ( ph -> O = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. (TopSet ` ndx ) , K >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   {ctp 4181   <.cop 4183   U.cuni 4436   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   ndxcnx 15854   Basecbs 15857   +g cplusg 15941  TopSetcts 15947   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ^ko cxko 21364   IIcii 22678   *pcpco 22800   Om1 comi 22801

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  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-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-topon 20716  df-om1 22806

This theorem is referenced by:  om1bas  22831  om1plusg  22834  om1tset  22835