Theorem pj1val 18108

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
pj1val	|- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) = ( iota_ x e. T E. y e. U X = ( x .+ y ) ) )

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

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

Proof of Theorem pj1val

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pj1fval.v	. . . 4 |- B = ( Base ` G )
2		pj1fval.a	. . . 4 |- .+ = ( +g ` G )
3		pj1fval.s	. . . 4 |- .(+) = ( LSSum ` G )
4		pj1fval.p	. . . 4 |- P = ( proj1 ` G )
5	1, 2, 3, 4	pj1fval 18107	. . 3 |- ( ( 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 ) ) ) )
6	5	adantr 481	. 2 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) -> ( T P U ) = ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) )
7		simpr 477	. . . . 5 |- ( ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) /\ z = X ) -> z = X )
8	7	eqeq1d 2624	. . . 4 |- ( ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) /\ z = X ) -> ( z = ( x .+ y ) <-> X = ( x .+ y ) ) )
9	8	rexbidv 3052	. . 3 |- ( ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) /\ z = X ) -> ( E. y e. U z = ( x .+ y ) <-> E. y e. U X = ( x .+ y ) ) )
10	9	riotabidv 6613	. 2 |- ( ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) /\ z = X ) -> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) = ( iota_ x e. T E. y e. U X = ( x .+ y ) ) )
11		simpr 477	. 2 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) -> X e. ( T .(+) U ) )
12		riotaex 6615	. . 3 |- ( iota_ x e. T E. y e. U X = ( x .+ y ) ) e. _V
13	12	a1i 11	. 2 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) -> ( iota_ x e. T E. y e. U X = ( x .+ y ) ) e. _V )
14	6, 10, 11, 13	fvmptd 6288	1 |- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) = ( iota_ x e. T E. y e. U X = ( x .+ y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   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:  pj1id  18112