Theorem eroprf2 6716

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 6715	1 |- ( ph -> .+^ : ( J X. J ) --> J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   E.wrex 2485   C_ wss 3166   class class class wbr 4044   X. cxp 4673   -->wf 5267  (class class class)co 5944   {coprab 5945   Er wer 6617   [cec 6618   /.cqs 6619

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-er 6620  df-ec 6622  df-qs 6626

This theorem is referenced by: (None)