Theorem oviec 6788

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 4798	. . . . 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 4798	. . . . 5 |- ( ( ( g e. S /\ h e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. g , h >. .~ <. t , s >. <-> ch ) )
9	6, 8	bi2anan9 608	. . . 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 6142	. . . . . 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 6142	. . . . . 6 |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. c , d >. .+ <. t , s >. ) = L )
17	13, 16	breqan12d 4099	. . . . 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 590	. . . 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 2296	. . . . . . 7 |- ( x e. Q <-> x e. ( ( S X. S ) /. .~ ) )
23	21	eleq2i 2296	. . . . . . 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 6059	. . . 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 2250	. . 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 6787	. 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 6142	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .+ <. C , D >. ) = H )
32	31	eceq1d 6716	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ H ] .~ )
33	28, 32	eqtrd 2262	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 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   <.cop 3669   class class class wbr 4083   {copab 4144   X. cxp 4717  (class class class)co 6001   {coprab 6002   Er wer 6677   [cec 6678   /.cqs 6679

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-er 6680  df-ec 6682  df-qs 6686

This theorem is referenced by:  addpipqqs  7557  mulpipqqs  7560