Theorem oviec 6586

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 4665	. . . . 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 4665	. . . . 5 |- ( ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. g , h >. .~ <. t , s >. <-> ch ) )
9	6, 8	bi2anan9 596	. . . 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 5957	. . . . . 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 5957	. . . . . 6 |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. c , d >. .+ <. t , s >. ) = L )
17	13, 16	breqan12d 3981	. . . . 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 578	. . . 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 2224	. . . . . . 7 |- ( x e. Q <-> x e. ( ( S X. S ) /. .~ ) )
23	21	eleq2i 2224	. . . . . . 7 |- ( y e. Q <-> y e. ( ( S X. S ) /. .~ ) )
24	22, 23	anbi12i 456	. . . . . 6 |- ( ( x e. Q /\ y e. Q ) <-> ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) )
25	24	anbi1i 454	. . . . 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 5876	. . . 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 2178	. . 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 6585	. 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 5957	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .+ <. C , D >. ) = H )
32	31	eceq1d 6516	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ H ] .~ )
33	28, 32	eqtrd 2190	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 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712   <.cop 3563   class class class wbr 3965   {copab 4024   X. cxp 4584  (class class class)co 5824   {coprab 5825   Er wer 6477   [cec 6478   /.cqs 6479

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fv 5178  df-ov 5827  df-oprab 5828  df-er 6480  df-ec 6482  df-qs 6486

This theorem is referenced by:  addpipqqs  7290  mulpipqqs  7293