Theorem th3q 6618

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 4643	. . . 4 |- ( ( A e. S /\ B e. S ) -> <. A , B >. e. ( S X. S ) )
2		th3q.1	. . . . 5 |- .~ e. _V
3	2	ecelqsi 6567	. . . 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 4643	. . . 4 |- ( ( C e. S /\ D e. S ) -> <. C , D >. e. ( S X. S ) )
6	2	ecelqsi 6567	. . . 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 336	. 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 2170	. . . 4 |- [ <. A , B >. ] .~ = [ <. A , B >. ] .~
10		eqid 2170	. . . 4 |- [ <. C , D >. ] .~ = [ <. C , D >. ] .~
11	9, 10	pm3.2i 270	. . 3 |- ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ )
12		eqid 2170	. . 3 |- [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~
13		opeq12 3767	. . . . . 6 |- ( ( w = A /\ v = B ) -> <. w , v >. = <. A , B >. )
14		eceq1 6548	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> [ <. w , v >. ] .~ = [ <. A , B >. ] .~ )
15	14	eqeq2d 2182	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ <-> [ <. A , B >. ] .~ = [ <. A , B >. ] .~ ) )
16	15	anbi1d 462	. . . . . . 7 |- ( <. w , v >. = <. A , B >. -> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. A , B >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) ) )
17		oveq1 5860	. . . . . . . . 9 |- ( <. w , v >. = <. A , B >. -> ( <. w , v >. .+ <. C , D >. ) = ( <. A , B >. .+ <. C , D >. ) )
18	17	eceq1d 6549	. . . . . . . 8 |- ( <. w , v >. = <. A , B >. -> [ ( <. w , v >. .+ <. C , D >. ) ] .~ = [ ( <. A , B >. .+ <. C , D >. ) ] .~ )
19	18	eqeq2d 2182	. . . . . . 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 470	. . . . . 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 2820	. . . 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 3767	. . . . . . 7 |- ( ( u = C /\ t = D ) -> <. u , t >. = <. C , D >. )
24		eceq1 6548	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> [ <. u , t >. ] .~ = [ <. C , D >. ] .~ )
25	24	eqeq2d 2182	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> ( [ <. C , D >. ] .~ = [ <. u , t >. ] .~ <-> [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) )
26	25	anbi2d 461	. . . . . . . 8 |- ( <. u , t >. = <. C , D >. -> ( ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. C , D >. ] .~ ) ) )
27		oveq2 5861	. . . . . . . . . 10 |- ( <. u , t >. = <. C , D >. -> ( <. w , v >. .+ <. u , t >. ) = ( <. w , v >. .+ <. C , D >. ) )
28	27	eceq1d 6549	. . . . . . . . 9 |- ( <. u , t >. = <. C , D >. -> [ ( <. w , v >. .+ <. u , t >. ) ] .~ = [ ( <. w , v >. .+ <. C , D >. ) ] .~ )
29	28	eqeq2d 2182	. . . . . . . 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 470	. . . . . . 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 2820	. . . . 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 1875	. . . 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 407	. . 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 430	. 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 6517	. . . 4 |- ( .~ e. _V -> [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V )
37	2, 36	ax-mp 5	. . 3 |- [ ( <. A , B >. .+ <. C , D >. ) ] .~ e. _V
38		eqeq1 2177	. . . . . . . 8 |- ( x = [ <. A , B >. ] .~ -> ( x = [ <. w , v >. ] .~ <-> [ <. A , B >. ] .~ = [ <. w , v >. ] .~ ) )
39		eqeq1 2177	. . . . . . . 8 |- ( y = [ <. C , D >. ] .~ -> ( y = [ <. u , t >. ] .~ <-> [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) )
40	38, 39	bi2anan9 601	. . . . . . 7 |- ( ( x = [ <. A , B >. ] .~ /\ y = [ <. C , D >. ] .~ ) -> ( ( x = [ <. w , v >. ] .~ /\ y = [ <. u , t >. ] .~ ) <-> ( [ <. A , B >. ] .~ = [ <. w , v >. ] .~ /\ [ <. C , D >. ] .~ = [ <. u , t >. ] .~ ) ) )
41		eqeq1 2177	. . . . . . 7 |- ( z = [ ( <. A , B >. .+ <. C , D >. ) ] .~ -> ( z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ <-> [ ( <. A , B >. .+ <. C , D >. ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
42	40, 41	bi2anan9 601	. . . . . 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 1189	. . . . 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 1863	. . . 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 6616	. . . 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 5974	. . 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 1321	. 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 103   <-> wb 104   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   <.cop 3586   class class class wbr 3989   X. cxp 4609  (class class class)co 5853   {coprab 5854   Er wer 6510   [cec 6511   /.cqs 6512

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by:  oviec  6619