Theorem pj1fval 16196

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 2986	. . . . 5 |- ( G e. V -> G e. _V )
3	2	3ad2ant1 1009	. . . 4 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> G e. _V )
4		fveq2 5696	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
5		pj1fval.v	. . . . . . . 8 |- B = ( Base ` G )
6	4, 5	syl6eqr 2493	. . . . . . 7 |- ( g = G -> ( Base ` g ) = B )
7	6	pweqd 3870	. . . . . 6 |- ( g = G -> ~P ( Base ` g ) = ~P B )
8		fveq2 5696	. . . . . . . . 9 |- ( g = G -> ( LSSum ` g ) = ( LSSum ` G ) )
9		pj1fval.s	. . . . . . . . 9 |- .(+) = ( LSSum ` G )
10	8, 9	syl6eqr 2493	. . . . . . . 8 |- ( g = G -> ( LSSum ` g ) = .(+) )
11	10	oveqd 6113	. . . . . . 7 |- ( g = G -> ( t ( LSSum ` g ) u ) = ( t .(+) u ) )
12		fveq2 5696	. . . . . . . . . . . 12 |- ( g = G -> ( +g ` g ) = ( +g ` G ) )
13		pj1fval.a	. . . . . . . . . . . 12 |- .+ = ( +g ` G )
14	12, 13	syl6eqr 2493	. . . . . . . . . . 11 |- ( g = G -> ( +g ` g ) = .+ )
15	14	oveqd 6113	. . . . . . . . . 10 |- ( g = G -> ( x ( +g ` g ) y ) = ( x .+ y ) )
16	15	eqeq2d 2454	. . . . . . . . 9 |- ( g = G -> ( z = ( x ( +g ` g ) y ) <-> z = ( x .+ y ) ) )
17	16	rexbidv 2741	. . . . . . . 8 |- ( g = G -> ( E. y e. u z = ( x ( +g ` g ) y ) <-> E. y e. u z = ( x .+ y ) ) )
18	17	riotabidv 6059	. . . . . . 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 4375	. . . . . 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 6153	. . . . 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 16141	. . . . 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 5706	. . . . . . . 8 |- ( Base ` G ) e. _V
23	5, 22	eqeltri 2513	. . . . . . 7 |- B e. _V
24	23	pwex 4480	. . . . . 6 |- ~P B e. _V
25	24, 24	mpt2ex 6655	. . . . 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 5779	. . . 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 2487	. 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 6105	. . . 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 2929	. . . 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 6060	. . 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 4375	. 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 989	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> T C_ B )
37	23	elpw2 4461	. . 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 990	. . 3 |- ( ( G e. V /\ T C_ B /\ U C_ B ) -> U C_ B )
40	23	elpw2 4461	. . 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 6121	. . . 4 |- ( T .(+) U ) e. _V
43	42	mptex 5953	. . 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 6223	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 965   = wceq 1369   e. wcel 1756   E.wrex 2721   _Vcvv 2977   C_ wss 3333   ~Pcpw 3865   e. cmpt 4355   ` cfv 5423   iota_crio 6056  (class class class)co 6096   e. cmpt2 6098   Basecbs 14179   +g cplusg 14243   LSSumclsm 16138   proj1cpj1 16139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-pj1 16141

This theorem is referenced by:  pj1val  16197  pj1f  16199