Theorem eroprf2 6797

Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
eropr2.1	|- J = ( A /. .~ )
eropr2.2	|- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. A ( ( x = [ p ] .~ /\ y = [ q ] .~ ) /\ z = [ ( p .+ q ) ] .~ ) }
eropr2.3	|- ( ph -> .~ e. X )
eropr2.4	|- ( ph -> .~ Er U )
eropr2.5	|- ( ph -> A C_ U )
eropr2.6	|- ( ph -> .+ : ( A X. A ) --> A )
eropr2.7	|- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. A /\ u e. A ) ) ) -> ( ( r .~ s /\ t .~ u ) -> ( r .+ t ) .~ ( s .+ u ) ) )

Assertion

Ref	Expression
eroprf2	|- ( ph -> .+^ : ( J X. J ) --> J )

Distinct variable groups:   q, p, r, s, t, u, x, y, z, A   X, p, q, r, s, t, u, z   .+ , p, q, r, s, t, u, x, y, z   .~ , p, q, r, s, t, u, x, y, z   J, p, q, x, y, z   ph, p, q, r, s, t, u, x, y, z

Allowed substitution hints:   .+^ ( x, y, z, u, t, s, r, q, p)   U( x, y, z, u, t, s, r, q, p)   J( u, t, s, r)   X( x, y)

Proof of Theorem eroprf2

Step	Hyp	Ref	Expression
1		eropr2.1	. 2 |- J = ( A /. .~ )
2		eropr2.3	. 2 |- ( ph -> .~ e. X )
3		eropr2.4	. 2 |- ( ph -> .~ Er U )
4		eropr2.5	. 2 |- ( ph -> A C_ U )
5		eropr2.6	. 2 |- ( ph -> .+ : ( A X. A ) --> A )
6		eropr2.7	. 2 |- ( ( ph /\ ( ( r e. A /\ s e. A ) /\ ( t e. A /\ u e. A ) ) ) -> ( ( r .~ s /\ t .~ u ) -> ( r .+ t ) .~ ( s .+ u ) ) )
7		eropr2.2	. 2 |- .+^ = { <. <. x , y >. , z >. | E. p e. A E. q e. A ( ( x = [ p ] .~ /\ y = [ q ] .~ ) /\ z = [ ( p .+ q ) ] .~ ) }
8	1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2, 1	eroprf 6796	1 |- ( ph -> .+^ : ( J X. J ) --> J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   class class class wbr 4088   X. cxp 4723   -->wf 5322  (class class class)co 6017   {coprab 6018   Er wer 6698   [cec 6699   /.cqs 6700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-er 6701  df-ec 6703  df-qs 6707

This theorem is referenced by: (None)