Theorem eroprf2 6631

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 6630	1 |- ( ph -> .+^ : ( J X. J ) --> J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   C_ wss 3131   class class class wbr 4005   X. cxp 4626   -->wf 5214  (class class class)co 5877   {coprab 5878   Er wer 6534   [cec 6535   /.cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-er 6537  df-ec 6539  df-qs 6543

This theorem is referenced by: (None)