Theorem aptap 8669

Description: Complex apartness (as defined at df-ap 8601) is a tight apartness (as defined at df-tap 7310). (Contributed by Jim Kingdon, 16-Feb-2025.)

Assertion

Ref	Expression
aptap	|- # TAp CC

Proof of Theorem aptap

Dummy variables q p r s t u v x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2200	. . . . . . . . . 10 |- ( u = ( 1st ` t ) -> ( u = ( p + ( _i x. q ) ) <-> ( 1st ` t ) = ( p + ( _i x. q ) ) ) )
2	1	anbi1d 465	. . . . . . . . 9 |- ( u = ( 1st ` t ) -> ( ( u = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) <-> ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) ) )
3	2	anbi1d 465	. . . . . . . 8 |- ( u = ( 1st ` t ) -> ( ( ( u = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) <-> ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) )
4	3	2rexbidv 2519	. . . . . . 7 |- ( u = ( 1st ` t ) -> ( E. r e. RR E. s e. RR ( ( u = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) <-> E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) )
5	4	2rexbidv 2519	. . . . . 6 |- ( u = ( 1st ` t ) -> ( E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( u = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) <-> E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) )
6		eqeq1 2200	. . . . . . . . . 10 |- ( v = ( 2nd ` t ) -> ( v = ( r + ( _i x. s ) ) <-> ( 2nd ` t ) = ( r + ( _i x. s ) ) ) )
7	6	anbi2d 464	. . . . . . . . 9 |- ( v = ( 2nd ` t ) -> ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) <-> ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) ) )
8	7	anbi1d 465	. . . . . . . 8 |- ( v = ( 2nd ` t ) -> ( ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) <-> ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) )
9	8	2rexbidv 2519	. . . . . . 7 |- ( v = ( 2nd ` t ) -> ( E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) <-> E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) )
10	9	2rexbidv 2519	. . . . . 6 |- ( v = ( 2nd ` t ) -> ( E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) <-> E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) )
11	5, 10	elopabi 6248	. . . . 5 |- ( t e. { <. u , v >. | E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( u = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) } -> E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) )
12		df-ap 8601	. . . . 5 |- # = { <. u , v >. | E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( u = ( p + ( _i x. q ) ) /\ v = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) }
13	11, 12	eleq2s 2288	. . . 4 |- ( t e. # -> E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) )
14	12	relopabi 4787	. . . . . . . . . 10 |- Rel #
15		simp-5l 543	. . . . . . . . . 10 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> t e. # )
16		1st2nd 6234	. . . . . . . . . 10 |- ( ( Rel # /\ t e. # ) -> t = <. ( 1st ` t ) , ( 2nd ` t ) >. )
17	14, 15, 16	sylancr 414	. . . . . . . . 9 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> t = <. ( 1st ` t ) , ( 2nd ` t ) >. )
18		simprll 537	. . . . . . . . . . 11 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( 1st ` t ) = ( p + ( _i x. q ) ) )
19		simp-5r 544	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> p e. RR )
20	19	recnd 8048	. . . . . . . . . . . 12 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> p e. CC )
21		ax-icn 7967	. . . . . . . . . . . . . 14 |- _i e. CC
22	21	a1i 9	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> _i e. CC )
23		simp-4r 542	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> q e. RR )
24	23	recnd 8048	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> q e. CC )
25	22, 24	mulcld 8040	. . . . . . . . . . . 12 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( _i x. q ) e. CC )
26	20, 25	addcld 8039	. . . . . . . . . . 11 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( p + ( _i x. q ) ) e. CC )
27	18, 26	eqeltrd 2270	. . . . . . . . . 10 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( 1st ` t ) e. CC )
28		simprlr 538	. . . . . . . . . . 11 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( 2nd ` t ) = ( r + ( _i x. s ) ) )
29		simpllr 534	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> r e. RR )
30	29	recnd 8048	. . . . . . . . . . . 12 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> r e. CC )
31		simplr 528	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> s e. RR )
32	31	recnd 8048	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> s e. CC )
33	22, 32	mulcld 8040	. . . . . . . . . . . 12 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( _i x. s ) e. CC )
34	30, 33	addcld 8039	. . . . . . . . . . 11 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( r + ( _i x. s ) ) e. CC )
35	28, 34	eqeltrd 2270	. . . . . . . . . 10 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( 2nd ` t ) e. CC )
36	27, 35	jca 306	. . . . . . . . 9 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> ( ( 1st ` t ) e. CC /\ ( 2nd ` t ) e. CC ) )
37		elxp6 6222	. . . . . . . . 9 |- ( t e. ( CC X. CC ) <-> ( t = <. ( 1st ` t ) , ( 2nd ` t ) >. /\ ( ( 1st ` t ) e. CC /\ ( 2nd ` t ) e. CC ) ) )
38	17, 36, 37	sylanbrc 417	. . . . . . . 8 |- ( ( ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) /\ s e. RR ) /\ ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) ) -> t e. ( CC X. CC ) )
39	38	rexlimdva2 2614	. . . . . . 7 |- ( ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) /\ r e. RR ) -> ( E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) -> t e. ( CC X. CC ) ) )
40	39	rexlimdva 2611	. . . . . 6 |- ( ( ( t e. # /\ p e. RR ) /\ q e. RR ) -> ( E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) -> t e. ( CC X. CC ) ) )
41	40	rexlimdva 2611	. . . . 5 |- ( ( t e. # /\ p e. RR ) -> ( E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) -> t e. ( CC X. CC ) ) )
42	41	rexlimdva 2611	. . . 4 |- ( t e. # -> ( E. p e. RR E. q e. RR E. r e. RR E. s e. RR ( ( ( 1st ` t ) = ( p + ( _i x. q ) ) /\ ( 2nd ` t ) = ( r + ( _i x. s ) ) ) /\ ( p #ℝ r \/ q #ℝ s ) ) -> t e. ( CC X. CC ) ) )
43	13, 42	mpd 13	. . 3 |- ( t e. # -> t e. ( CC X. CC ) )
44	43	ssriv 3183	. 2 |- # C_ ( CC X. CC )
45		apirr 8624	. . . 4 |- ( x e. CC -> -. x # x )
46	45	rgen 2547	. . 3 |- A. x e. CC -. x # x
47		apsym 8625	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x # y <-> y # x ) )
48	47	biimpd 144	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x # y -> y # x ) )
49	48	rgen2 2580	. . 3 |- A. x e. CC A. y e. CC ( x # y -> y # x )
50	46, 49	pm3.2i 272	. 2 |- ( A. x e. CC -. x # x /\ A. x e. CC A. y e. CC ( x # y -> y # x ) )
51		apcotr 8626	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x # y -> ( x # z \/ y # z ) ) )
52	51	rgen3 2581	. . 3 |- A. x e. CC A. y e. CC A. z e. CC ( x # y -> ( x # z \/ y # z ) )
53		apti 8641	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x = y <-> -. x # y ) )
54	53	biimprd 158	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( -. x # y -> x = y ) )
55	54	rgen2 2580	. . 3 |- A. x e. CC A. y e. CC ( -. x # y -> x = y )
56	52, 55	pm3.2i 272	. 2 |- ( A. x e. CC A. y e. CC A. z e. CC ( x # y -> ( x # z \/ y # z ) ) /\ A. x e. CC A. y e. CC ( -. x # y -> x = y ) )
57		dftap2 7311	. 2 |- ( # TAp CC <-> ( # C_ ( CC X. CC ) /\ ( A. x e. CC -. x # x /\ A. x e. CC A. y e. CC ( x # y -> y # x ) ) /\ ( A. x e. CC A. y e. CC A. z e. CC ( x # y -> ( x # z \/ y # z ) ) /\ A. x e. CC A. y e. CC ( -. x # y -> x = y ) ) ) )
58	44, 50, 56, 57	mpbir3an 1181	1 |- # TAp CC

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   C_ wss 3153   <.cop 3621   class class class wbr 4029   {copab 4089   X. cxp 4657   Rel wrel 4664   ` cfv 5254  (class class class)co 5918   1stc1st 6191   2ndc2nd 6192   TAp wtap 7309   CCcc 7870   RRcr 7871   _ici 7874   + caddc 7875   x. cmul 7877   #_ℝ creap 8593   # cap 8600

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pap 7308  df-tap 7310  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601

This theorem is referenced by: (None)