Theorem th3qlem1 6639

Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
th3qlem1.1	|- .~ Er S
th3qlem1.3	|- ( ( ( y e. S /\ w e. S ) /\ ( z e. S /\ v e. S ) ) -> ( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) )

Assertion

Ref	Expression
th3qlem1	|- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> E* x E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) )

Distinct variable groups:   x, y, z, w, v, .+   x, .~ , y, z, w, v   x, S, y, z, w, v   x, A, y, z, w, v   x, B, y, z, w, v

Proof of Theorem th3qlem1

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ee4anv 1934	. . . 4 |- ( E. y E. z E. w E. v ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) <-> ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ E. w E. v ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) )
2		an4 586	. . . . . . 7 |- ( ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) <-> ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) /\ ( x = [ ( y .+ z ) ] .~ /\ u = [ ( w .+ v ) ] .~ ) ) )
3		eleq1 2240	. . . . . . . . . . . . 13 |- ( A = [ y ] .~ -> ( A e. ( S /. .~ ) <-> [ y ] .~ e. ( S /. .~ ) ) )
4		eleq1 2240	. . . . . . . . . . . . 13 |- ( B = [ z ] .~ -> ( B e. ( S /. .~ ) <-> [ z ] .~ e. ( S /. .~ ) ) )
5	3, 4	bi2anan9 606	. . . . . . . . . . . 12 |- ( ( A = [ y ] .~ /\ B = [ z ] .~ ) -> ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) <-> ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) ) )
6	5	adantr 276	. . . . . . . . . . 11 |- ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) -> ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) <-> ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) ) )
7	6	biimpac 298	. . . . . . . . . 10 |- ( ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) /\ ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) ) -> ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) )
8		eqtr2 2196	. . . . . . . . . . . . 13 |- ( ( A = [ y ] .~ /\ A = [ w ] .~ ) -> [ y ] .~ = [ w ] .~ )
9		eqtr2 2196	. . . . . . . . . . . . 13 |- ( ( B = [ z ] .~ /\ B = [ v ] .~ ) -> [ z ] .~ = [ v ] .~ )
10	8, 9	anim12i 338	. . . . . . . . . . . 12 |- ( ( ( A = [ y ] .~ /\ A = [ w ] .~ ) /\ ( B = [ z ] .~ /\ B = [ v ] .~ ) ) -> ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) )
11	10	an4s 588	. . . . . . . . . . 11 |- ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) -> ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) )
12	11	adantl 277	. . . . . . . . . 10 |- ( ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) /\ ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) ) -> ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) )
13		th3qlem1.1	. . . . . . . . . . . 12 |- .~ Er S
14	13	a1i 9	. . . . . . . . . . 11 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> .~ Er S )
15		simprl 529	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ y ] .~ = [ w ] .~ )
16		erdm 6547	. . . . . . . . . . . . . . . 16 |- ( .~ Er S -> dom .~ = S )
17	13, 16	ax-mp 5	. . . . . . . . . . . . . . 15 |- dom .~ = S
18		simpll 527	. . . . . . . . . . . . . . 15 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ y ] .~ e. ( S /. .~ ) )
19		ecelqsdm 6607	. . . . . . . . . . . . . . 15 |- ( ( dom .~ = S /\ [ y ] .~ e. ( S /. .~ ) ) -> y e. S )
20	17, 18, 19	sylancr 414	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> y e. S )
21	14, 20	erth 6581	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( y .~ w <-> [ y ] .~ = [ w ] .~ ) )
22	15, 21	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> y .~ w )
23		simprr 531	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ z ] .~ = [ v ] .~ )
24		simplr 528	. . . . . . . . . . . . . . 15 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ z ] .~ e. ( S /. .~ ) )
25		ecelqsdm 6607	. . . . . . . . . . . . . . 15 |- ( ( dom .~ = S /\ [ z ] .~ e. ( S /. .~ ) ) -> z e. S )
26	17, 24, 25	sylancr 414	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> z e. S )
27	14, 26	erth 6581	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( z .~ v <-> [ z ] .~ = [ v ] .~ ) )
28	23, 27	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> z .~ v )
29	15, 18	eqeltrrd 2255	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ w ] .~ e. ( S /. .~ ) )
30		ecelqsdm 6607	. . . . . . . . . . . . . 14 |- ( ( dom .~ = S /\ [ w ] .~ e. ( S /. .~ ) ) -> w e. S )
31	17, 29, 30	sylancr 414	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> w e. S )
32	23, 24	eqeltrrd 2255	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ v ] .~ e. ( S /. .~ ) )
33		ecelqsdm 6607	. . . . . . . . . . . . . 14 |- ( ( dom .~ = S /\ [ v ] .~ e. ( S /. .~ ) ) -> v e. S )
34	17, 32, 33	sylancr 414	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> v e. S )
35		th3qlem1.3	. . . . . . . . . . . . 13 |- ( ( ( y e. S /\ w e. S ) /\ ( z e. S /\ v e. S ) ) -> ( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) )
36	20, 31, 26, 34, 35	syl22anc 1239	. . . . . . . . . . . 12 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) )
37	22, 28, 36	mp2and 433	. . . . . . . . . . 11 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( y .+ z ) .~ ( w .+ v ) )
38	14, 37	erthi 6583	. . . . . . . . . 10 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ )
39	7, 12, 38	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) /\ ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) ) -> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ )
40		eqeq12 2190	. . . . . . . . 9 |- ( ( x = [ ( y .+ z ) ] .~ /\ u = [ ( w .+ v ) ] .~ ) -> ( x = u <-> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ ) )
41	39, 40	syl5ibrcom 157	. . . . . . . 8 |- ( ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) /\ ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) ) -> ( ( x = [ ( y .+ z ) ] .~ /\ u = [ ( w .+ v ) ] .~ ) -> x = u ) )
42	41	expimpd 363	. . . . . . 7 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> ( ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) /\ ( x = [ ( y .+ z ) ] .~ /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
43	2, 42	biimtrid 152	. . . . . 6 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> ( ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
44	43	exlimdvv 1897	. . . . 5 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> ( E. w E. v ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
45	44	exlimdvv 1897	. . . 4 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> ( E. y E. z E. w E. v ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
46	1, 45	biimtrrid 153	. . 3 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> ( ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ E. w E. v ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
47	46	alrimivv 1875	. 2 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> A. x A. u ( ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ E. w E. v ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
48		eqeq1 2184	. . . . . 6 |- ( x = u -> ( x = [ ( y .+ z ) ] .~ <-> u = [ ( y .+ z ) ] .~ ) )
49	48	anbi2d 464	. . . . 5 |- ( x = u -> ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) <-> ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ u = [ ( y .+ z ) ] .~ ) ) )
50	49	2exbidv 1868	. . . 4 |- ( x = u -> ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) <-> E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ u = [ ( y .+ z ) ] .~ ) ) )
51		eceq1 6572	. . . . . . . 8 |- ( y = w -> [ y ] .~ = [ w ] .~ )
52	51	eqeq2d 2189	. . . . . . 7 |- ( y = w -> ( A = [ y ] .~ <-> A = [ w ] .~ ) )
53		eceq1 6572	. . . . . . . 8 |- ( z = v -> [ z ] .~ = [ v ] .~ )
54	53	eqeq2d 2189	. . . . . . 7 |- ( z = v -> ( B = [ z ] .~ <-> B = [ v ] .~ ) )
55	52, 54	bi2anan9 606	. . . . . 6 |- ( ( y = w /\ z = v ) -> ( ( A = [ y ] .~ /\ B = [ z ] .~ ) <-> ( A = [ w ] .~ /\ B = [ v ] .~ ) ) )
56		oveq12 5886	. . . . . . . 8 |- ( ( y = w /\ z = v ) -> ( y .+ z ) = ( w .+ v ) )
57	56	eceq1d 6573	. . . . . . 7 |- ( ( y = w /\ z = v ) -> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ )
58	57	eqeq2d 2189	. . . . . 6 |- ( ( y = w /\ z = v ) -> ( u = [ ( y .+ z ) ] .~ <-> u = [ ( w .+ v ) ] .~ ) )
59	55, 58	anbi12d 473	. . . . 5 |- ( ( y = w /\ z = v ) -> ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ u = [ ( y .+ z ) ] .~ ) <-> ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) )
60	59	cbvex2v 1924	. . . 4 |- ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ u = [ ( y .+ z ) ] .~ ) <-> E. w E. v ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) )
61	50, 60	bitrdi 196	. . 3 |- ( x = u -> ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) <-> E. w E. v ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) )
62	61	mo4 2087	. 2 |- ( E* x E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) <-> A. x A. u ( ( E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) /\ E. w E. v ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) -> x = u ) )
63	47, 62	sylibr 134	1 |- ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) -> E* x E. y E. z ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   E*wmo 2027   e. wcel 2148   class class class wbr 4005   dom cdm 4628  (class class class)co 5877   Er wer 6534   [cec 6535   /.cqs 6536

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fv 5226  df-ov 5880  df-er 6537  df-ec 6539  df-qs 6543

This theorem is referenced by:  th3qlem2  6640