Theorem oviec 6535

Description: Express an operation on equivalence classes of ordered pairs in terms of equivalence class of operations on ordered pairs. See iset.mm for additional comments describing the hypotheses. (Unnecessary distinct variable restrictions were removed by David Abernethy, 4-Jun-2013.) (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 4-Jun-2013.)

Hypotheses

Ref	Expression
oviec.1	|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> H e. ( S X. S ) )
oviec.2	|- ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) -> K e. ( S X. S ) )
oviec.3	|- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> L e. ( S X. S ) )
oviec.4	|- .~ e. _V
oviec.5	|- .~ Er ( S X. S )
oviec.7	|- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) }
oviec.8	|- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ph <-> ps ) )
oviec.9	|- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ph <-> ch ) )
oviec.10	|- .+ = { <. <. x , y >. , z >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = J ) ) }
oviec.11	|- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> J = K )
oviec.12	|- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> J = L )
oviec.13	|- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> J = H )
oviec.14	|- .+^ = { <. <. x , y >. , z >. | ( ( x e. Q /\ y e. Q ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) }
oviec.15	|- Q = ( ( S X. S ) /. .~ )
oviec.16	|- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\ ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( ps /\ ch ) -> K .~ L ) )

Assertion

Ref	Expression
oviec	|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) = [ H ] .~ )

Distinct variable groups:   a, b, c, d, f, u, v, w, x, y, z, C   D, a, b, c, d, f, u, v, w, x, y, z   x, J, y, z   g, a, h, A, b, c, d, f, u, v, w, x, y, z   ch, u, v, w, z   f, H, u, v, w, x, y, z   B, a, b, c, d, f, g, h, u, v, w, x, y, z   f, K, u, v, w, x, y, z   ps, u, v, w, z   f, L, u, v, w, x, y, z   ph, x, y   s, a, t, S, b, c, d, f, g, h, u, v, w, x, y, z   .+ , a, b, c, d, g, h, s, t, x, y, z   .~ , a, b, c, d, g, h, s, t, x, y, z

Allowed substitution hints:   ph( z, w, v, u, t, f, g, h, s, a, b, c, d)   ps( x, y, t, f, g, h, s, a, b, c, d)   ch( x, y, t, f, g, h, s, a, b, c, d)   A( t, s)   B( t, s)   C( t, g, h, s)   D( t, g, h, s)   .+ ( w, v, u, f)   .+^ ( x, y, z, w, v, u, t, f, g, h, s, a, b, c, d)   Q( x, y, z, w, v, u, t, f, g, h, s, a, b, c, d)   .~ ( w, v, u, f)   H( t, g, h, s, a, b, c, d)   J( w, v, u, t, f, g, h, s, a, b, c, d)   K( t, g, h, s, a, b, c, d)   L( t, g, h, s, a, b, c, d)

Proof of Theorem oviec

Step	Hyp	Ref	Expression
1		oviec.4	. . 3 |- .~ e. _V
2		oviec.5	. . 3 |- .~ Er ( S X. S )
3		oviec.16	. . . 4 |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\ ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( ps /\ ch ) -> K .~ L ) )
4		oviec.8	. . . . . 6 |- ( ( ( z = a /\ w = b ) /\ ( v = c /\ u = d ) ) -> ( ph <-> ps ) )
5		oviec.7	. . . . . 6 |- .~ = { <. x , y >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ph ) ) }
6	4, 5	opbrop 4618	. . . . 5 |- ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) -> ( <. a , b >. .~ <. c , d >. <-> ps ) )
7		oviec.9	. . . . . 6 |- ( ( ( z = g /\ w = h ) /\ ( v = t /\ u = s ) ) -> ( ph <-> ch ) )
8	7, 5	opbrop 4618	. . . . 5 |- ( ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. g , h >. .~ <. t , s >. <-> ch ) )
9	6, 8	bi2anan9 595	. . . 4 |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\ ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( <. a , b >. .~ <. c , d >. /\ <. g , h >. .~ <. t , s >. ) <-> ( ps /\ ch ) ) )
10		oviec.2	. . . . . . 7 |- ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) -> K e. ( S X. S ) )
11		oviec.11	. . . . . . 7 |- ( ( ( w = a /\ v = b ) /\ ( u = g /\ f = h ) ) -> J = K )
12		oviec.10	. . . . . . 7 |- .+ = { <. <. x , y >. , z >. | ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = J ) ) }
13	10, 11, 12	ovi3 5907	. . . . . 6 |- ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) -> ( <. a , b >. .+ <. g , h >. ) = K )
14		oviec.3	. . . . . . 7 |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> L e. ( S X. S ) )
15		oviec.12	. . . . . . 7 |- ( ( ( w = c /\ v = d ) /\ ( u = t /\ f = s ) ) -> J = L )
16	14, 15, 12	ovi3 5907	. . . . . 6 |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. c , d >. .+ <. t , s >. ) = L )
17	13, 16	breqan12d 3945	. . . . 5 |- ( ( ( ( a e. S /\ b e. S ) /\ ( g e. S /\ h e. S ) ) /\ ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( <. a , b >. .+ <. g , h >. ) .~ ( <. c , d >. .+ <. t , s >. ) <-> K .~ L ) )
18	17	an4s 577	. . . 4 |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\ ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( <. a , b >. .+ <. g , h >. ) .~ ( <. c , d >. .+ <. t , s >. ) <-> K .~ L ) )
19	3, 9, 18	3imtr4d 202	. . 3 |- ( ( ( ( a e. S /\ b e. S ) /\ ( c e. S /\ d e. S ) ) /\ ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) ) -> ( ( <. a , b >. .~ <. c , d >. /\ <. g , h >. .~ <. t , s >. ) -> ( <. a , b >. .+ <. g , h >. ) .~ ( <. c , d >. .+ <. t , s >. ) ) )
20		oviec.14	. . . 4 |- .+^ = { <. <. x , y >. , z >. | ( ( x e. Q /\ y e. Q ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) }
21		oviec.15	. . . . . . . 8 |- Q = ( ( S X. S ) /. .~ )
22	21	eleq2i 2206	. . . . . . 7 |- ( x e. Q <-> x e. ( ( S X. S ) /. .~ ) )
23	21	eleq2i 2206	. . . . . . 7 |- ( y e. Q <-> y e. ( ( S X. S ) /. .~ ) )
24	22, 23	anbi12i 455	. . . . . 6 |- ( ( x e. Q /\ y e. Q ) <-> ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) )
25	24	anbi1i 453	. . . . 5 |- ( ( ( x e. Q /\ y e. Q ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) <-> ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) )
26	25	oprabbii 5826	. . . 4 |- { <. <. x , y >. , z >. | ( ( x e. Q /\ y e. Q ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) }
27	20, 26	eqtri 2160	. . 3 |- .+^ = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. a E. b E. c E. d ( ( x = [ <. a , b >. ] .~ /\ y = [ <. c , d >. ] .~ ) /\ z = [ ( <. a , b >. .+ <. c , d >. ) ] .~ ) ) }
28	1, 2, 19, 27	th3q 6534	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )
29		oviec.1	. . . 4 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> H e. ( S X. S ) )
30		oviec.13	. . . 4 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> J = H )
31	29, 30, 12	ovi3 5907	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .+ <. C , D >. ) = H )
32	31	eceq1d 6465	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ H ] .~ )
33	28, 32	eqtrd 2172	1 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) = [ H ] .~ )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686   <.cop 3530   class class class wbr 3929   {copab 3988   X. cxp 4537  (class class class)co 5774   {coprab 5775   Er wer 6426   [cec 6427   /.cqs 6428

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-er 6429  df-ec 6431  df-qs 6435

This theorem is referenced by:  addpipqqs  7178  mulpipqqs  7181