Theorem pj1fval 18107

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 3212	. . . . 5 |- ( G e. V -> G e. _V )
3	2	3ad2ant1 1082	. . . 4 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> G e. _V )
4		fveq2 6191	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5		pj1fval.v	. . . . . . . 8 |- B = ( Base ` G )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( g = G -> ( Base ` g ) = B )
7	6	pweqd 4163	. . . . . 6 |- ( g = G -> ~P ( Base ` g ) = ~P B )
8		fveq2 6191	. . . . . . . . 9 |- ( g = G -> ( LSSum ` g ) = ( LSSum ` G ) )
9		pj1fval.s	. . . . . . . . 9 |- .(+) = ( LSSum ` G )
10	8, 9	syl6eqr 2674	. . . . . . . 8 |- ( g = G -> ( LSSum ` g ) = .(+) )
11	10	oveqd 6667	. . . . . . 7 |- ( g = G -> ( t ( LSSum ` g ) u ) = ( t .(+) u ) )
12		fveq2 6191	. . . . . . . . . . . 12 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
13		pj1fval.a	. . . . . . . . . . . 12 |- .+ = ( +g ` G )
14	12, 13	syl6eqr 2674	. . . . . . . . . . 11 |- ( g = G -> ( +g ` g ) = .+ )
15	14	oveqd 6667	. . . . . . . . . 10 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
16	15	eqeq2d 2632	. . . . . . . . 9 |- ( g = G -> ( z = ( x ( +g ` g ) y ) <-> z = ( x .+ y ) ) )
17	16	rexbidv 3052	. . . . . . . 8 |- ( g = G -> ( E. y e. u z = ( x ( +g ` g ) y ) <-> E. y e. u z = ( x .+ y ) ) )
18	17	riotabidv 6613	. . . . . . 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 4733	. . . . . 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 6717	. . . . 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 18052	. . . . 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 6201	. . . . . . . 8 |- ( Base ` G ) e. _V
23	5, 22	eqeltri 2697	. . . . . . 7 |- B e. _V
24	23	pwex 4848	. . . . . 6 |- ~P B e. _V
25	24, 24	mpt2ex 7247	. . . . 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 6282	. . . 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 2668	. 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 6659	. . . 4 |- ( ( t = T /\ u = U ) -> ( t .(+) u ) = ( T .(+) U ) )
30	29	adantl 482	. . 3 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> ( t .(+) u ) = ( T .(+) U ) )
31		simprl 794	. . . 4 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> t = T )
32		simprr 796	. . . . 5 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( t = T /\ u = U ) ) -> u = U )
33	32	rexeqdv 3145	. . . 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 6614	. . 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 4733	. 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 1062	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T C_ B )
37	23	elpw2 4828	. . 3 |- ( T e. ~P B <-> T C_ B )
38	36, 37	sylibr 224	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T e. ~P B )
39		simp3 1063	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U C_ B )
40	23	elpw2 4828	. . 3 |- ( U e. ~P B <-> U C_ B )
41	39, 40	sylibr 224	. 2 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U e. ~P B )
42		ovex 6678	. . . 4 |- ( T .(+) U ) e. _V
43	42	mptex 6486	. . 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 6788	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941   LSSumclsm 18049   proj1cpj1 18050

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

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-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  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-1st 7168  df-2nd 7169  df-pj1 18052

This theorem is referenced by:  pj1val  18108  pj1f  18110