Theorem th3qlem2 6232

Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

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 >. ) ) )

Assertion

Ref	Expression
th3qlem2	|- ( ( A e. ( ( S X. S ) /. .~ ) /\ B e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )

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

Allowed substitution hints:   A( g, h)   B( g, h)

Proof of Theorem th3qlem2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		th3q.2	. . 3 |- .~ Er ( S X. S )
2		eqid 2081	. . . . 5 |- ( S X. S ) = ( S X. S )
3		breq1 3788	. . . . . . . 8 |- ( <. w , v >. = s -> ( <. w , v >. .~ <. u , t >. <-> s .~ <. u , t >. ) )
4	3	anbi1d 452	. . . . . . 7 |- ( <. w , v >. = s -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) <-> ( s .~ <. u , t >. /\ x .~ y ) ) )
5		oveq1 5539	. . . . . . . 8 |- ( <. w , v >. = s -> ( <. w , v >. .+ x ) = ( s .+ x ) )
6	5	breq1d 3795	. . . . . . 7 |- ( <. w , v >. = s -> ( ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) <-> ( s .+ x ) .~ ( <. u , t >. .+ y ) ) )
7	4, 6	imbi12d 232	. . . . . 6 |- ( <. w , v >. = s -> ( ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) <-> ( ( s .~ <. u , t >. /\ x .~ y ) -> ( s .+ x ) .~ ( <. u , t >. .+ y ) ) ) )
8	7	imbi2d 228	. . . . 5 |- ( <. w , v >. = s -> ( ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) ) <-> ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( s .~ <. u , t >. /\ x .~ y ) -> ( s .+ x ) .~ ( <. u , t >. .+ y ) ) ) ) )
9		breq2 3789	. . . . . . . 8 |- ( <. u , t >. = f -> ( s .~ <. u , t >. <-> s .~ f ) )
10	9	anbi1d 452	. . . . . . 7 |- ( <. u , t >. = f -> ( ( s .~ <. u , t >. /\ x .~ y ) <-> ( s .~ f /\ x .~ y ) ) )
11		oveq1 5539	. . . . . . . 8 |- ( <. u , t >. = f -> ( <. u , t >. .+ y ) = ( f .+ y ) )
12	11	breq2d 3797	. . . . . . 7 |- ( <. u , t >. = f -> ( ( s .+ x ) .~ ( <. u , t >. .+ y ) <-> ( s .+ x ) .~ ( f .+ y ) ) )
13	10, 12	imbi12d 232	. . . . . 6 |- ( <. u , t >. = f -> ( ( ( s .~ <. u , t >. /\ x .~ y ) -> ( s .+ x ) .~ ( <. u , t >. .+ y ) ) <-> ( ( s .~ f /\ x .~ y ) -> ( s .+ x ) .~ ( f .+ y ) ) ) )
14	13	imbi2d 228	. . . . 5 |- ( <. u , t >. = f -> ( ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( s .~ <. u , t >. /\ x .~ y ) -> ( s .+ x ) .~ ( <. u , t >. .+ y ) ) ) <-> ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( s .~ f /\ x .~ y ) -> ( s .+ x ) .~ ( f .+ y ) ) ) ) )
15		breq1 3788	. . . . . . . . . 10 |- ( <. s , f >. = x -> ( <. s , f >. .~ <. g , h >. <-> x .~ <. g , h >. ) )
16	15	anbi2d 451	. . . . . . . . 9 |- ( <. s , f >. = x -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) <-> ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) ) )
17		oveq2 5540	. . . . . . . . . 10 |- ( <. s , f >. = x -> ( <. w , v >. .+ <. s , f >. ) = ( <. w , v >. .+ x ) )
18	17	breq1d 3795	. . . . . . . . 9 |- ( <. s , f >. = x -> ( ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) <-> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
19	16, 18	imbi12d 232	. . . . . . . 8 |- ( <. s , f >. = x -> ( ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) <-> ( ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) )
20	19	imbi2d 228	. . . . . . 7 |- ( <. s , f >. = x -> ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) <-> ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) ) )
21		breq2 3789	. . . . . . . . . 10 |- ( <. g , h >. = y -> ( x .~ <. g , h >. <-> x .~ y ) )
22	21	anbi2d 451	. . . . . . . . 9 |- ( <. g , h >. = y -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) <-> ( <. w , v >. .~ <. u , t >. /\ x .~ y ) ) )
23		oveq2 5540	. . . . . . . . . 10 |- ( <. g , h >. = y -> ( <. u , t >. .+ <. g , h >. ) = ( <. u , t >. .+ y ) )
24	23	breq2d 3797	. . . . . . . . 9 |- ( <. g , h >. = y -> ( ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) <-> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) )
25	22, 24	imbi12d 232	. . . . . . . 8 |- ( <. g , h >. = y -> ( ( ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) ) <-> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) ) )
26	25	imbi2d 228	. . . . . . 7 |- ( <. g , h >. = y -> ( ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) <-> ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) ) ) )
27		th3q.4	. . . . . . . 8 |- ( ( ( ( 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 >. ) ) )
28	27	expcom 114	. . . . . . 7 |- ( ( ( s e. S /\ f e. S ) /\ ( g e. S /\ h e. S ) ) -> ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) -> ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) ) ) )
29	2, 20, 26, 28	2optocl 4435	. . . . . 6 |- ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) ) )
30	29	com12 30	. . . . 5 |- ( ( ( w e. S /\ v e. S ) /\ ( u e. S /\ t e. S ) ) -> ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) -> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) ) )
31	2, 8, 14, 30	2optocl 4435	. . . 4 |- ( ( s e. ( S X. S ) /\ f e. ( S X. S ) ) -> ( ( x e. ( S X. S ) /\ y e. ( S X. S ) ) -> ( ( s .~ f /\ x .~ y ) -> ( s .+ x ) .~ ( f .+ y ) ) ) )
32	31	imp 122	. . 3 |- ( ( ( s e. ( S X. S ) /\ f e. ( S X. S ) ) /\ ( x e. ( S X. S ) /\ y e. ( S X. S ) ) ) -> ( ( s .~ f /\ x .~ y ) -> ( s .+ x ) .~ ( f .+ y ) ) )
33	1, 32	th3qlem1 6231	. 2 |- ( ( A e. ( ( S X. S ) /. .~ ) /\ B e. ( ( S X. S ) /. .~ ) ) -> E* z E. s E. x ( ( A = [ s ] .~ /\ B = [ x ] .~ ) /\ z = [ ( s .+ x ) ] .~ ) )
34		vex 2604	. . . . . . 7 |- w e. _V
35		vex 2604	. . . . . . 7 |- v e. _V
36	34, 35	opex 3984	. . . . . 6 |- <. w , v >. e. _V
37		vex 2604	. . . . . . 7 |- u e. _V
38		vex 2604	. . . . . . 7 |- t e. _V
39	37, 38	opex 3984	. . . . . 6 |- <. u , t >. e. _V
40		eceq1 6164	. . . . . . . . 9 |- ( s = <. w , v >. -> [ s ] .~ = [ <. w , v >. ] .~ )
41	40	eqeq2d 2092	. . . . . . . 8 |- ( s = <. w , v >. -> ( A = [ s ] .~ <-> A = [ <. w , v >. ] .~ ) )
42		eceq1 6164	. . . . . . . . 9 |- ( x = <. u , t >. -> [ x ] .~ = [ <. u , t >. ] .~ )
43	42	eqeq2d 2092	. . . . . . . 8 |- ( x = <. u , t >. -> ( B = [ x ] .~ <-> B = [ <. u , t >. ] .~ ) )
44	41, 43	bi2anan9 570	. . . . . . 7 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( ( A = [ s ] .~ /\ B = [ x ] .~ ) <-> ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) ) )
45		oveq12 5541	. . . . . . . . 9 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( s .+ x ) = ( <. w , v >. .+ <. u , t >. ) )
46	45	eceq1d 6165	. . . . . . . 8 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> [ ( s .+ x ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ )
47	46	eqeq2d 2092	. . . . . . 7 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( z = [ ( s .+ x ) ] .~ <-> z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
48	44, 47	anbi12d 456	. . . . . 6 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( ( ( A = [ s ] .~ /\ B = [ x ] .~ ) /\ z = [ ( s .+ x ) ] .~ ) <-> ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) ) )
49	36, 39, 48	spc2ev 2693	. . . . 5 |- ( ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> E. s E. x ( ( A = [ s ] .~ /\ B = [ x ] .~ ) /\ z = [ ( s .+ x ) ] .~ ) )
50	49	exlimivv 1817	. . . 4 |- ( E. u E. t ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> E. s E. x ( ( A = [ s ] .~ /\ B = [ x ] .~ ) /\ z = [ ( s .+ x ) ] .~ ) )
51	50	exlimivv 1817	. . 3 |- ( E. w E. v E. u E. t ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) -> E. s E. x ( ( A = [ s ] .~ /\ B = [ x ] .~ ) /\ z = [ ( s .+ x ) ] .~ ) )
52	51	moimi 2006	. 2 |- ( E* z E. s E. x ( ( A = [ s ] .~ /\ B = [ x ] .~ ) /\ z = [ ( s .+ x ) ] .~ ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
53	33, 52	syl 14	1 |- ( ( A e. ( ( S X. S ) /. .~ ) /\ B e. ( ( S X. S ) /. .~ ) ) -> E* z E. w E. v E. u E. t ( ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) /\ z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   E*wmo 1942   _Vcvv 2601   <.cop 3401   class class class wbr 3785   X. cxp 4361  (class class class)co 5532   Er wer 6126   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fv 4930  df-ov 5535  df-er 6129  df-ec 6131  df-qs 6135

This theorem is referenced by:  th3qcor  6233  th3q  6234