Theorem pj1fval 16562

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 3127	. . . . 5 |- ( G e. V -> G e. _V )
3	2	3ad2ant1 1017	. . . 4 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> G e. _V )
4		fveq2 5871	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5		pj1fval.v	. . . . . . . 8 |- B = ( Base ` G )
6	4, 5	syl6eqr 2526	. . . . . . 7 |- ( g = G -> ( Base ` g ) = B )
7	6	pweqd 4020	. . . . . 6 |- ( g = G -> ~P ( Base ` g ) = ~P B )
8		fveq2 5871	. . . . . . . . 9 |- ( g = G -> ( LSSum ` g ) = ( LSSum ` G ) )
9		pj1fval.s	. . . . . . . . 9 |- .(+) = ( LSSum ` G )
10	8, 9	syl6eqr 2526	. . . . . . . 8 |- ( g = G -> ( LSSum ` g ) = .(+) )
11	10	oveqd 6311	. . . . . . 7 |- ( g = G -> ( t ( LSSum ` g ) u ) = ( t .(+) u ) )
12		fveq2 5871	. . . . . . . . . . . 12 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
13		pj1fval.a	. . . . . . . . . . . 12 |- .+ = ( +g ` G )
14	12, 13	syl6eqr 2526	. . . . . . . . . . 11 |- ( g = G -> ( +g ` g ) = .+ )
15	14	oveqd 6311	. . . . . . . . . 10 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
16	15	eqeq2d 2481	. . . . . . . . 9 |- ( g = G -> ( z = ( x ( +g ` g ) y ) <-> z = ( x .+ y ) ) )
17	16	rexbidv 2978	. . . . . . . 8 |- ( g = G -> ( E. y e. u z = ( x ( +g ` g ) y ) <-> E. y e. u z = ( x .+ y ) ) )
18	17	riotabidv 6257	. . . . . . 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 4530	. . . . . 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 6353	. . . . 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 16507	. . . . 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 5881	. . . . . . . 8 |- ( Base ` G ) e. _V
23	5, 22	eqeltri 2551	. . . . . . 7 |- B e. _V
24	23	pwex 4635	. . . . . 6 |- ~P B e. _V
25	24, 24	mpt2ex 6870	. . . . 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 5956	. . . 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 16	. . 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 2520	. 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 6303	. . . 4 |- ( ( t = T /\ u = U ) -> ( t .(+) u ) = ( T .(+) U ) )
30	29	adantl 466	. . 3 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> ( t .(+) u ) = ( T .(+) U ) )
31		simprl 755	. . . 4 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> t = T )
32		simprr 756	. . . . 5 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> u = U )
33	32	rexeqdv 3070	. . . 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 6258	. . 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 4530	. 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 997	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T C_ B )
37	23	elpw2 4616	. . 3 |- ( T e. ~P B <-> T C_ B )
38	36, 37	sylibr 212	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T e. ~P B )
39		simp3 998	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U C_ B )
40	23	elpw2 4616	. . 3 |- ( U e. ~P B <-> U C_ B )
41	39, 40	sylibr 212	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U e. ~P B )
42		ovex 6319	. . . 4 |- ( T .(+) U ) e. _V
43	42	mptex 6141	. . 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 6424	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 369   /\ w3a 973   = wceq 1379   e. wcel 1767   E.wrex 2818   _Vcvv 3118   C_ wss 3481   ~Pcpw 4015   |-> cmpt 4510   ` cfv 5593   iota_crio 6254  (class class class)co 6294   |-> cmpt2 6296   Basecbs 14502   +g cplusg 14567   LSSumclsm 16504   proj1cpj1 16505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-pj1 16507

This theorem is referenced by:  pj1val  16563  pj1f  16565