Theorem oviec 6640

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 4705	. . . . 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 4705	. . . . 5 |- ( ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. g , h >. .~ <. t , s >. <-> ch ) )
9	6, 8	bi2anan9 606	. . . 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 6010	. . . . . 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 6010	. . . . . 6 |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. c , d >. .+ <. t , s >. ) = L )
17	13, 16	breqan12d 4019	. . . . 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 588	. . . 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 203	. . 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 2244	. . . . . . 7 |- ( x e. Q <-> x e. ( ( S X. S ) /. .~ ) )
23	21	eleq2i 2244	. . . . . . 7 |- ( y e. Q <-> y e. ( ( S X. S ) /. .~ ) )
24	22, 23	anbi12i 460	. . . . . 6 |- ( ( x e. Q /\ y e. Q ) <-> ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) )
25	24	anbi1i 458	. . . . 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 5929	. . . 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 2198	. . 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 6639	. 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 6010	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .+ <. C , D >. ) = H )
32	31	eceq1d 6570	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ H ] .~ )
33	28, 32	eqtrd 2210	1 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ .+^ [ <. C , D >. ] .~ ) = [ H ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   <.cop 3595   class class class wbr 4003   {copab 4063   X. cxp 4624  (class class class)co 5874   {coprab 5875   Er wer 6531   [cec 6532   /.cqs 6533

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-in1 614  ax-in2 615  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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-er 6534  df-ec 6536  df-qs 6540

This theorem is referenced by:  addpipqqs  7368  mulpipqqs  7371