Theorem th3qlem2 6725

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 2205	. . . . 5 |- ( S X. S ) = ( S X. S )
3		breq1 4047	. . . . . . . 8 |- ( <. w , v >. = s -> ( <. w , v >. .~ <. u , t >. <-> s .~ <. u , t >. ) )
4	3	anbi1d 465	. . . . . . 7 |- ( <. w , v >. = s -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ y ) <-> ( s .~ <. u , t >. /\ x .~ y ) ) )
5		oveq1 5951	. . . . . . . 8 |- ( <. w , v >. = s -> ( <. w , v >. .+ x ) = ( s .+ x ) )
6	5	breq1d 4054	. . . . . . 7 |- ( <. w , v >. = s -> ( ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) <-> ( s .+ x ) .~ ( <. u , t >. .+ y ) ) )
7	4, 6	imbi12d 234	. . . . . 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 230	. . . . 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 4048	. . . . . . . 8 |- ( <. u , t >. = f -> ( s .~ <. u , t >. <-> s .~ f ) )
10	9	anbi1d 465	. . . . . . 7 |- ( <. u , t >. = f -> ( ( s .~ <. u , t >. /\ x .~ y ) <-> ( s .~ f /\ x .~ y ) ) )
11		oveq1 5951	. . . . . . . 8 |- ( <. u , t >. = f -> ( <. u , t >. .+ y ) = ( f .+ y ) )
12	11	breq2d 4056	. . . . . . 7 |- ( <. u , t >. = f -> ( ( s .+ x ) .~ ( <. u , t >. .+ y ) <-> ( s .+ x ) .~ ( f .+ y ) ) )
13	10, 12	imbi12d 234	. . . . . 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 230	. . . . 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 4047	. . . . . . . . . 10 |- ( <. s , f >. = x -> ( <. s , f >. .~ <. g , h >. <-> x .~ <. g , h >. ) )
16	15	anbi2d 464	. . . . . . . . 9 |- ( <. s , f >. = x -> ( ( <. w , v >. .~ <. u , t >. /\ <. s , f >. .~ <. g , h >. ) <-> ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) ) )
17		oveq2 5952	. . . . . . . . . 10 |- ( <. s , f >. = x -> ( <. w , v >. .+ <. s , f >. ) = ( <. w , v >. .+ x ) )
18	17	breq1d 4054	. . . . . . . . 9 |- ( <. s , f >. = x -> ( ( <. w , v >. .+ <. s , f >. ) .~ ( <. u , t >. .+ <. g , h >. ) <-> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) ) )
19	16, 18	imbi12d 234	. . . . . . . 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 230	. . . . . . 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 4048	. . . . . . . . . 10 |- ( <. g , h >. = y -> ( x .~ <. g , h >. <-> x .~ y ) )
22	21	anbi2d 464	. . . . . . . . 9 |- ( <. g , h >. = y -> ( ( <. w , v >. .~ <. u , t >. /\ x .~ <. g , h >. ) <-> ( <. w , v >. .~ <. u , t >. /\ x .~ y ) ) )
23		oveq2 5952	. . . . . . . . . 10 |- ( <. g , h >. = y -> ( <. u , t >. .+ <. g , h >. ) = ( <. u , t >. .+ y ) )
24	23	breq2d 4056	. . . . . . . . 9 |- ( <. g , h >. = y -> ( ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ <. g , h >. ) <-> ( <. w , v >. .+ x ) .~ ( <. u , t >. .+ y ) ) )
25	22, 24	imbi12d 234	. . . . . . . 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 230	. . . . . . 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 116	. . . . . . 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 4752	. . . . . 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 4752	. . . 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 124	. . 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 6724	. 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 2775	. . . . . . 7 |- w e. _V
35		vex 2775	. . . . . . 7 |- v e. _V
36	34, 35	opex 4273	. . . . . 6 |- <. w , v >. e. _V
37		vex 2775	. . . . . . 7 |- u e. _V
38		vex 2775	. . . . . . 7 |- t e. _V
39	37, 38	opex 4273	. . . . . 6 |- <. u , t >. e. _V
40		eceq1 6655	. . . . . . . . 9 |- ( s = <. w , v >. -> [ s ] .~ = [ <. w , v >. ] .~ )
41	40	eqeq2d 2217	. . . . . . . 8 |- ( s = <. w , v >. -> ( A = [ s ] .~ <-> A = [ <. w , v >. ] .~ ) )
42		eceq1 6655	. . . . . . . . 9 |- ( x = <. u , t >. -> [ x ] .~ = [ <. u , t >. ] .~ )
43	42	eqeq2d 2217	. . . . . . . 8 |- ( x = <. u , t >. -> ( B = [ x ] .~ <-> B = [ <. u , t >. ] .~ ) )
44	41, 43	bi2anan9 606	. . . . . . 7 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( ( A = [ s ] .~ /\ B = [ x ] .~ ) <-> ( A = [ <. w , v >. ] .~ /\ B = [ <. u , t >. ] .~ ) ) )
45		oveq12 5953	. . . . . . . . 9 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( s .+ x ) = ( <. w , v >. .+ <. u , t >. ) )
46	45	eceq1d 6656	. . . . . . . 8 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> [ ( s .+ x ) ] .~ = [ ( <. w , v >. .+ <. u , t >. ) ] .~ )
47	46	eqeq2d 2217	. . . . . . 7 |- ( ( s = <. w , v >. /\ x = <. u , t >. ) -> ( z = [ ( s .+ x ) ] .~ <-> z = [ ( <. w , v >. .+ <. u , t >. ) ] .~ ) )
48	44, 47	anbi12d 473	. . . . . 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 2869	. . . . 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 1920	. . . 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 1920	. . 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 2119	. 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 104   = wceq 1373   E.wex 1515   E*wmo 2055   e. wcel 2176   _Vcvv 2772   <.cop 3636   class class class wbr 4044   X. cxp 4673  (class class class)co 5944   Er wer 6617   [cec 6618   /.cqs 6619

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fv 5279  df-ov 5947  df-er 6620  df-ec 6622  df-qs 6626

This theorem is referenced by:  th3qcor  6726  th3q  6727