Theorem th3qlem1 6615

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 1927	. . . 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 581	. . . . . . 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 2233	. . . . . . . . . . . . 13 |- ( A = [ y ] .~ -> ( A e. ( S /. .~ ) <-> [ y ] .~ e. ( S /. .~ ) ) )
4		eleq1 2233	. . . . . . . . . . . . 13 |- ( B = [ z ] .~ -> ( B e. ( S /. .~ ) <-> [ z ] .~ e. ( S /. .~ ) ) )
5	3, 4	bi2anan9 601	. . . . . . . . . . . 12 |- ( ( A = [ y ] .~ /\ B = [ z ] .~ ) -> ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) <-> ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) ) )
6	5	adantr 274	. . . . . . . . . . 11 |- ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) -> ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) <-> ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) ) )
7	6	biimpac 296	. . . . . . . . . 10 |- ( ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) /\ ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) ) -> ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) )
8		eqtr2 2189	. . . . . . . . . . . . 13 |- ( ( A = [ y ] .~ /\ A = [ w ] .~ ) -> [ y ] .~ = [ w ] .~ )
9		eqtr2 2189	. . . . . . . . . . . . 13 |- ( ( B = [ z ] .~ /\ B = [ v ] .~ ) -> [ z ] .~ = [ v ] .~ )
10	8, 9	anim12i 336	. . . . . . . . . . . 12 |- ( ( ( A = [ y ] .~ /\ A = [ w ] .~ ) /\ ( B = [ z ] .~ /\ B = [ v ] .~ ) ) -> ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) )
11	10	an4s 583	. . . . . . . . . . 11 |- ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) -> ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) )
12	11	adantl 275	. . . . . . . . . 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 526	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ y ] .~ = [ w ] .~ )
16		erdm 6523	. . . . . . . . . . . . . . . 16 |- ( .~ Er S -> dom .~ = S )
17	13, 16	ax-mp 5	. . . . . . . . . . . . . . 15 |- dom .~ = S
18		simpll 524	. . . . . . . . . . . . . . 15 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ y ] .~ e. ( S /. .~ ) )
19		ecelqsdm 6583	. . . . . . . . . . . . . . 15 |- ( ( dom .~ = S /\ [ y ] .~ e. ( S /. .~ ) ) -> y e. S )
20	17, 18, 19	sylancr 412	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> y e. S )
21	14, 20	erth 6557	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( y .~ w <-> [ y ] .~ = [ w ] .~ ) )
22	15, 21	mpbird 166	. . . . . . . . . . . 12 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> y .~ w )
23		simprr 527	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ z ] .~ = [ v ] .~ )
24		simplr 525	. . . . . . . . . . . . . . 15 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ z ] .~ e. ( S /. .~ ) )
25		ecelqsdm 6583	. . . . . . . . . . . . . . 15 |- ( ( dom .~ = S /\ [ z ] .~ e. ( S /. .~ ) ) -> z e. S )
26	17, 24, 25	sylancr 412	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> z e. S )
27	14, 26	erth 6557	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( z .~ v <-> [ z ] .~ = [ v ] .~ ) )
28	23, 27	mpbird 166	. . . . . . . . . . . 12 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> z .~ v )
29	15, 18	eqeltrrd 2248	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ w ] .~ e. ( S /. .~ ) )
30		ecelqsdm 6583	. . . . . . . . . . . . . 14 |- ( ( dom .~ = S /\ [ w ] .~ e. ( S /. .~ ) ) -> w e. S )
31	17, 29, 30	sylancr 412	. . . . . . . . . . . . 13 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> w e. S )
32	23, 24	eqeltrrd 2248	. . . . . . . . . . . . . 14 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ v ] .~ e. ( S /. .~ ) )
33		ecelqsdm 6583	. . . . . . . . . . . . . 14 |- ( ( dom .~ = S /\ [ v ] .~ e. ( S /. .~ ) ) -> v e. S )
34	17, 32, 33	sylancr 412	. . . . . . . . . . . . 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 1234	. . . . . . . . . . . 12 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( ( y .~ w /\ z .~ v ) -> ( y .+ z ) .~ ( w .+ v ) ) )
37	22, 28, 36	mp2and 431	. . . . . . . . . . 11 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> ( y .+ z ) .~ ( w .+ v ) )
38	14, 37	erthi 6559	. . . . . . . . . 10 |- ( ( ( [ y ] .~ e. ( S /. .~ ) /\ [ z ] .~ e. ( S /. .~ ) ) /\ ( [ y ] .~ = [ w ] .~ /\ [ z ] .~ = [ v ] .~ ) ) -> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ )
39	7, 12, 38	syl2anc 409	. . . . . . . . 9 |- ( ( ( A e. ( S /. .~ ) /\ B e. ( S /. .~ ) ) /\ ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ ( A = [ w ] .~ /\ B = [ v ] .~ ) ) ) -> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ )
40		eqeq12 2183	. . . . . . . . 9 |- ( ( x = [ ( y .+ z ) ] .~ /\ u = [ ( w .+ v ) ] .~ ) -> ( x = u <-> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ ) )
41	39, 40	syl5ibrcom 156	. . . . . . . 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 361	. . . . . . 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	syl5bi 151	. . . . . 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 1890	. . . . 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 1890	. . . 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	syl5bir 152	. . 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 1868	. 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 2177	. . . . . 6 |- ( x = u -> ( x = [ ( y .+ z ) ] .~ <-> u = [ ( y .+ z ) ] .~ ) )
49	48	anbi2d 461	. . . . 5 |- ( x = u -> ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ x = [ ( y .+ z ) ] .~ ) <-> ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ u = [ ( y .+ z ) ] .~ ) ) )
50	49	2exbidv 1861	. . . 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 6548	. . . . . . . 8 |- ( y = w -> [ y ] .~ = [ w ] .~ )
52	51	eqeq2d 2182	. . . . . . 7 |- ( y = w -> ( A = [ y ] .~ <-> A = [ w ] .~ ) )
53		eceq1 6548	. . . . . . . 8 |- ( z = v -> [ z ] .~ = [ v ] .~ )
54	53	eqeq2d 2182	. . . . . . 7 |- ( z = v -> ( B = [ z ] .~ <-> B = [ v ] .~ ) )
55	52, 54	bi2anan9 601	. . . . . 6 |- ( ( y = w /\ z = v ) -> ( ( A = [ y ] .~ /\ B = [ z ] .~ ) <-> ( A = [ w ] .~ /\ B = [ v ] .~ ) ) )
56		oveq12 5862	. . . . . . . 8 |- ( ( y = w /\ z = v ) -> ( y .+ z ) = ( w .+ v ) )
57	56	eceq1d 6549	. . . . . . 7 |- ( ( y = w /\ z = v ) -> [ ( y .+ z ) ] .~ = [ ( w .+ v ) ] .~ )
58	57	eqeq2d 2182	. . . . . 6 |- ( ( y = w /\ z = v ) -> ( u = [ ( y .+ z ) ] .~ <-> u = [ ( w .+ v ) ] .~ ) )
59	55, 58	anbi12d 470	. . . . 5 |- ( ( y = w /\ z = v ) -> ( ( ( A = [ y ] .~ /\ B = [ z ] .~ ) /\ u = [ ( y .+ z ) ] .~ ) <-> ( ( A = [ w ] .~ /\ B = [ v ] .~ ) /\ u = [ ( w .+ v ) ] .~ ) ) )
60	59	cbvex2v 1917	. . . 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 195	. . 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 2080	. 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 133	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 103   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   E*wmo 2020   e. wcel 2141   class class class wbr 3989   dom cdm 4611  (class class class)co 5853   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-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

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-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-fv 5206  df-ov 5856  df-er 6513  df-ec 6515  df-qs 6519

This theorem is referenced by:  th3qlem2  6616