Theorem oviec 6697

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 4739	. . . . 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 4739	. . . . 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 6057	. . . . . 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 6057	. . . . . 6 |- ( ( ( c e. S /\ d e. S ) /\ ( t e. S /\ s e. S ) ) -> ( <. c , d >. .+ <. t , s >. ) = L )
17	13, 16	breqan12d 4046	. . . . 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 2260	. . . . . . 7 |- ( x e. Q <-> x e. ( ( S X. S ) /. .~ ) )
23	21	eleq2i 2260	. . . . . . 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 5974	. . . 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 2214	. . 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 6696	. 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 6057	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( <. A , B >. .+ <. C , D >. ) = H )
32	31	eceq1d 6625	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ H ] .~ )
33	28, 32	eqtrd 2226	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 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   <.cop 3622   class class class wbr 4030   {copab 4090   X. cxp 4658  (class class class)co 5919   {coprab 5920   Er wer 6586   [cec 6587   /.cqs 6588

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-er 6589  df-ec 6591  df-qs 6595

This theorem is referenced by:  addpipqqs  7432  mulpipqqs  7435