Theorem aptap 8694

Description: Complex apartness (as defined at df-ap 8626) is a tight apartness (as defined at df-tap 7333). (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 2203	. . . . . . . . . 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 2522	. . . . . . 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 2522	. . . . . 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 2203	. . . . . . . . . 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 2522	. . . . . . 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 2522	. . . . . 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 6262	. . . . 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 8626	. . . . 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 2291	. . . 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 4792	. . . . . . . . . 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 6248	. . . . . . . . . 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 8072	. . . . . . . . . . . 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 7991	. . . . . . . . . . . . . 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 8072	. . . . . . . . . . . . 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 8064	. . . . . . . . . . . 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 8063	. . . . . . . . . . 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 2273	. . . . . . . . . 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 8072	. . . . . . . . . . . 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 8072	. . . . . . . . . . . . 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 8064	. . . . . . . . . . . 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 8063	. . . . . . . . . . 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 2273	. . . . . . . . . 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 6236	. . . . . . . . 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 2617	. . . . . . 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 2614	. . . . . 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 2614	. . . . 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 2614	. . . 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 3188	. 2 |- # C_ ( CC X. CC )
45		apirr 8649	. . . 4 |- ( x e. CC -> -. x # x )
46	45	rgen 2550	. . 3 |- A. x e. CC -. x # x
47		apsym 8650	. . . . 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 2583	. . 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 8651	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x # y -> ( x # z \/ y # z ) ) )
52	51	rgen3 2584	. . 3 |- A. x e. CC A. y e. CC A. z e. CC ( x # y -> ( x # z \/ y # z ) )
53		apti 8666	. . . . 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 2583	. . 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 7334	. 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 2167   A.wral 2475   E.wrex 2476   C_ wss 3157   <.cop 3626   class class class wbr 4034   {copab 4094   X. cxp 4662   Rel wrel 4669   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   TAp wtap 7332   CCcc 7894   RRcr 7895   _ici 7898   + caddc 7899   x. cmul 7901   #_ℝ creap 8618   # cap 8625

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pap 7331  df-tap 7333  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626

This theorem is referenced by: (None)