Theorem th3q 6694

Description: Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
th3q.1	|- .~ e. _V
th3q.2	|- .~ Er ( S X. S )
th3q.4	|- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
th3q.5	|- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }

Assertion

Ref	Expression
th3q	|- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )

Distinct variable groups:   x, y, z, w, v, u, t, s, f, g, h, .~   x, S, y, z, w, v, u, t, s, f, g, h   x, A, y, z, w, v, u, t, s, f   x, B, y, z, w, v, u, t, s, f   x, C, y, z, w, v, u, t   x, D, y, z, w, v, u, t   x, .+ , y, z, w, v, u, t, s, f, g, h

Allowed substitution hints:   A( g, h)   B( g, h)   C( f, g, h, s)   D( f, g, h, s)   G( x, y, z, w, v, u, t, f, g, h, s)

Proof of Theorem th3q

Step	Hyp	Ref	Expression
1		opelxpi 4691	. . . 4 |- ( ( A e. S /\ B e. S ) -> <. A , B >. e. ( S X. S ) )
2		th3q.1	. . . . 5 |- .~ e. _V
3	2	ecelqsi 6643	. . . 4 |- ( <. A , B >. e. ( S X. S ) -> [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) )
4	1, 3	syl 14	. . 3 |- ( ( A e. S /\ B e. S ) -> [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) )
5		opelxpi 4691	. . . 4 |- ( ( C e. S /\ D e. S ) -> <. C , D >. e. ( S X. S ) )
6	2	ecelqsi 6643	. . . 4 |- ( <. C , D >. e. ( S X. S ) -> [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) )
7	5, 6	syl 14	. . 3 |- ( ( C e. S /\ D e. S ) -> [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) )
8	4, 7	anim12i 338	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) ) )
9		eqid 2193	. . . 4 |- [ <. A , B >. ] .~ = [ <. A , B >. ] .~
10		eqid 2193	. . . 4 |- [ <. C , D >. ] .~ = [ <. C , D >. ] .~
11	9, 10	pm3.2i 272	. . 3 |- ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ )
12		eqid 2193	. . 3 |- [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~
13		opeq12 3806	. . . . . 6 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14		eceq1 6622	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> [ <. w , v >. ] .~ = [ <. A , B >. ] .~ )
15	14	eqeq2d 2205	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ <-> [ <. A , B >. ] .~ = [ <. A , B >. ] .~ ) )
16	15	anbi1d 465	. . . . . . 7 |- ( <. w , v >. = <. A , B >. -> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) ) )
17		oveq1 5925	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> ( <. w , v >. .+ <. C , D >. ) = ( <. A , B >. .+ <. C , D >. ) )
18	17	eceq1d 6623	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> [ ( <. w , v >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )
19	18	eqeq2d 2205	. . . . . . 7 |- ( <. w , v >. = <. A , B >. -> ( [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) )
20	16, 19	anbi12d 473	. . . . . 6 |- ( <. w , v >. = <. A , B >. -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) ) )
21	13, 20	syl 14	. . . . 5 |- ( ( w = A /\ v = B ) -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) ) )
22	21	spc2egv 2850	. . . 4 |- ( ( A e. S /\ B e. S ) -> ( ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> E. w E. v ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) ) )
23		opeq12 3806	. . . . . . 7 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
24		eceq1 6622	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> [ <. u , t >. ] .~ = [ <. C , D >. ] .~ )
25	24	eqeq2d 2205	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> ( [ <. C , D >. ] .~ = [ <. u , t >. ] .~ <-> [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) )
26	25	anbi2d 464	. . . . . . . 8 |- ( <. u , t >. = <. C , D >. -> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) ) )
27		oveq2 5926	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> ( <. w , v >. .+ <. u , t >. ) = ( <. w , v >. .+ <. C , D >. ) )
28	27	eceq1d 6623	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> [ ( <. w , v >. .+ <. u , t >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ )
29	28	eqeq2d 2205	. . . . . . . 8 |- ( <. u , t >. = <. C , D >. -> ( [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) )
30	26, 29	anbi12d 473	. . . . . . 7 |- ( <. u , t >. = <. C , D >. -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) ) )
31	23, 30	syl 14	. . . . . 6 |- ( ( u = C /\ t = D ) -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) ) )
32	31	spc2egv 2850	. . . . 5 |- ( ( C e. S /\ D e. S ) -> ( ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) -> E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
33	32	2eximdv 1893	. . . 4 |- ( ( C e. S /\ D e. S ) -> ( E. w E. v ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
34	22, 33	sylan9 409	. . 3 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( ( ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
35	11, 12, 34	mp2ani 432	. 2 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
36		ecexg 6591	. . . 4 |- ( .~ e. _V -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V )
37	2, 36	ax-mp 5	. . 3 |- [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V
38		eqeq1 2200	. . . . . . . 8 |- ( x = [ <. A , B >. ] .~ -> ( x = [ <. w , v >. ] .~ <-> [ <. A , B >. ] .~ = [ <. w , v >. ] .~ ) )
39		eqeq1 2200	. . . . . . . 8 |- ( y = [ <. C , D >. ] .~ -> ( y = [ <. u , t >. ] .~ <-> [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) )
40	38, 39	bi2anan9 606	. . . . . . 7 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ ) -> ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) ) )
41		eqeq1 2200	. . . . . . 7 |- ( z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ -> ( z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
42	40, 41	bi2anan9 606	. . . . . 6 |- ( ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ ) /\ z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> ( ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
43	42	3impa 1196	. . . . 5 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ /\ z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> ( ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
44	43	4exbidv 1881	. . . 4 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ /\ z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) -> ( E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) <-> E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
45		th3q.2	. . . . 5 |- .~ Er ( S X. S )
46		th3q.4	. . . . 5 |- ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) /\ ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
47	2, 45, 46	th3qlem2 6692	. . . 4 |- ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
48		th3q.5	. . . 4 |- G = { <. <. x , y >. , z >. | ( ( x e. ( ( S X. S ) /. .~ ) /\ y e. ( ( S X. S ) /. .~ ) ) /\ E. w E. v E. u E. t ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) }
49	44, 47, 48	ovig 6040	. . 3 |- ( ( [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) )
50	37, 49	mp3an3 1337	. 2 |- ( ( [ <. A , B >. ] .~ e. ( ( S X. S ) /. .~ ) /\ [ <. C , D >. ] .~ e. ( ( S X. S ) /. .~ ) ) -> ( E. w E. v E. u E. t ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) /\ [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ ) )
51	8, 35, 50	sylc 62	1 |- ( ( ( A e. S /\ B e. S ) /\ ( C e. S /\ D e. S ) ) -> ( [ <. A , B >. ] .~ G [ <. C , D >. ] .~ ) = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   <.cop 3621   class class class wbr 4029   X. cxp 4657  (class class class)co 5918   {coprab 5919   Er wer 6584   [cec 6585   /.cqs 6586

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-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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  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-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-er 6587  df-ec 6589  df-qs 6593

This theorem is referenced by:  oviec  6695