Theorem pj1fval 17392

Description: The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
pj1fval.v	|- B = ( Base ` G )
pj1fval.a	|- .+ = ( +g ` G )
pj1fval.s	|- .(+) = ( LSSum ` G )
pj1fval.p	|- P = ( proj1 ` G )

Assertion

Ref	Expression
pj1fval	|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( T P U ) = ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) )

Distinct variable groups:   z, .+   x, y, z, B   x, T, y, z   x, U, y, z   x, .(+) , y, z   x, G, y, z   x, V, y, z

Allowed substitution hints:   P( x, y, z)   .+ ( x, y)

Proof of Theorem pj1fval

Dummy variables t g u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pj1fval.p	. . 3 |- P = ( proj1 ` G )
2		elex 3065	. . . . 5 |- ( G e. V -> G e. _V )
3	2	3ad2ant1 1035	. . . 4 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> G e. _V )
4		fveq2 5887	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5		pj1fval.v	. . . . . . . 8 |- B = ( Base ` G )
6	4, 5	syl6eqr 2513	. . . . . . 7 |- ( g = G -> ( Base ` g ) = B )
7	6	pweqd 3967	. . . . . 6 |- ( g = G -> ~P ( Base ` g ) = ~P B )
8		fveq2 5887	. . . . . . . . 9 |- ( g = G -> ( LSSum ` g ) = ( LSSum ` G ) )
9		pj1fval.s	. . . . . . . . 9 |- .(+) = ( LSSum ` G )
10	8, 9	syl6eqr 2513	. . . . . . . 8 |- ( g = G -> ( LSSum ` g ) = .(+) )
11	10	oveqd 6331	. . . . . . 7 |- ( g = G -> ( t ( LSSum ` g ) u ) = ( t .(+) u ) )
12		fveq2 5887	. . . . . . . . . . . 12 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
13		pj1fval.a	. . . . . . . . . . . 12 |- .+ = ( +g ` G )
14	12, 13	syl6eqr 2513	. . . . . . . . . . 11 |- ( g = G -> ( +g ` g ) = .+ )
15	14	oveqd 6331	. . . . . . . . . 10 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
16	15	eqeq2d 2471	. . . . . . . . 9 |- ( g = G -> ( z = ( x ( +g ` g ) y ) <-> z = ( x .+ y ) ) )
17	16	rexbidv 2912	. . . . . . . 8 |- ( g = G -> ( E. y e. u z = ( x ( +g ` g ) y ) <-> E. y e. u z = ( x .+ y ) ) )
18	17	riotabidv 6278	. . . . . . 7 |- ( g = G -> ( iota_ x e. t E. y e. u z = ( x ( +g ` g ) y ) ) = ( iota_ x e. t E. y e. u z = ( x .+ y ) ) )
19	11, 18	mpteq12dv 4494	. . . . . 6 |- ( g = G -> ( z e. ( t ( LSSum ` g ) u ) |-> ( iota_ x e. t E. y e. u z = ( x ( +g ` g ) y ) ) ) = ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) )
20	7, 7, 19	mpt2eq123dv 6379	. . . . 5 |- ( g = G -> ( t e. ~P ( Base ` g ) , u e. ~P ( Base ` g ) |-> ( z e. ( t ( LSSum ` g ) u ) |-> ( iota_ x e. t E. y e. u z = ( x ( +g ` g ) y ) ) ) ) = ( t e. ~P B , u e. ~P B |-> ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) ) )
21		df-pj1 17337	. . . . 5 |- proj1 = ( g e. _V |-> ( t e. ~P ( Base ` g ) , u e. ~P ( Base ` g ) |-> ( z e. ( t ( LSSum ` g ) u ) |-> ( iota_ x e. t E. y e. u z = ( x ( +g ` g ) y ) ) ) ) )
22		fvex 5897	. . . . . . . 8 |- ( Base ` G ) e. _V
23	5, 22	eqeltri 2535	. . . . . . 7 |- B e. _V
24	23	pwex 4599	. . . . . 6 |- ~P B e. _V
25	24, 24	mpt2ex 6896	. . . . 5 |- ( t e. ~P B , u e. ~P B |-> ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) ) e. _V
26	20, 21, 25	fvmpt 5970	. . . 4 |- ( G e. _V -> ( proj1 ` G ) = ( t e. ~P B , u e. ~P B |-> ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) ) )
27	3, 26	syl 17	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( proj1 ` G ) = ( t e. ~P B , u e. ~P B |-> ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) ) )
28	1, 27	syl5eq 2507	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> P = ( t e. ~P B , u e. ~P B |-> ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) ) )
29		oveq12 6323	. . . 4 |- ( ( t = T /\ u = U ) -> ( t .(+) u ) = ( T .(+) U ) )
30	29	adantl 472	. . 3 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> ( t .(+) u ) = ( T .(+) U ) )
31		simprl 769	. . . 4 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> t = T )
32		simprr 771	. . . . 5 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> u = U )
33	32	rexeqdv 3005	. . . 4 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> ( E. y e. u z = ( x .+ y ) <-> E. y e. U z = ( x .+ y ) ) )
34	31, 33	riotaeqbidv 6279	. . 3 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) = ( iota_ x e. T E. y e. U z = ( x .+ y ) ) )
35	30, 34	mpteq12dv 4494	. 2 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> ( z e. ( t .(+) u ) |-> ( iota_ x e. t E. y e. u z = ( x .+ y ) ) ) = ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) )
36		simp2 1015	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T C_ B )
37	23	elpw2 4580	. . 3 |- ( T e. ~P B <-> T C_ B )
38	36, 37	sylibr 217	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T e. ~P B )
39		simp3 1016	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U C_ B )
40	23	elpw2 4580	. . 3 |- ( U e. ~P B <-> U C_ B )
41	39, 40	sylibr 217	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U e. ~P B )
42		ovex 6342	. . . 4 |- ( T .(+) U ) e. _V
43	42	mptex 6160	. . 3 |- ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) e. _V
44	43	a1i 11	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) e. _V )
45	28, 35, 38, 41, 44	ovmpt2d 6450	1 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( T P U ) = ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) )

Syntax hints:   -> wi 4   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   E.wrex 2749   _Vcvv 3056   C_ wss 3415   ~Pcpw 3962   |-> cmpt 4474   ` cfv 5600   iota_crio 6275  (class class class)co 6314   |-> cmpt2 6316   Basecbs 15169   +g cplusg 15238   LSSumclsm 17334   proj1cpj1 17335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-pj1 17337

This theorem is referenced by:  pj1val  17393  pj1f  17395