Theorem eroprf2 6688

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 6687	1 |- ( ph -> .+^ : ( J X. J ) --> J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   class class class wbr 4033   X. cxp 4661   -->wf 5254  (class class class)co 5922   {coprab 5923   Er wer 6589   [cec 6590   /.cqs 6591

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-er 6592  df-ec 6594  df-qs 6598

This theorem is referenced by: (None)