Theorem eroprf2 6607

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 6606	1 |- ( ph -> .+^ : ( J X. J ) --> J )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   C_ wss 3121   class class class wbr 3989   X. cxp 4609   -->wf 5194  (class class class)co 5853   {coprab 5854   Er wer 6510   [cec 6511   /.cqs 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by: (None)