Theorem aptap 8758

Description: Complex apartness (as defined at df-ap 8690) is a tight apartness (as defined at df-tap 7397). (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 2214	. . . . . . . . . 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 2533	. . . . . . 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 2533	. . . . . 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 2214	. . . . . . . . . 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 2533	. . . . . . 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 2533	. . . . . 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 6304	. . . . 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 8690	. . . . 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 2302	. . . 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 4821	. . . . . . . . . 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 6290	. . . . . . . . . 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 8136	. . . . . . . . . . . 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 8055	. . . . . . . . . . . . . 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 8136	. . . . . . . . . . . . 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 8128	. . . . . . . . . . . 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 8127	. . . . . . . . . . 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 2284	. . . . . . . . . 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 8136	. . . . . . . . . . . 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 8136	. . . . . . . . . . . . 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 8128	. . . . . . . . . . . 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 8127	. . . . . . . . . . 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 2284	. . . . . . . . . 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 6278	. . . . . . . . 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 2628	. . . . . . 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 2625	. . . . . 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 2625	. . . . 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 2625	. . . 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 3205	. 2 |- # C_ ( CC X. CC )
45		apirr 8713	. . . 4 |- ( x e. CC -> -. x # x )
46	45	rgen 2561	. . 3 |- A. x e. CC -. x # x
47		apsym 8714	. . . . 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 2594	. . 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 8715	. . . 4 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x # y -> ( x # z \/ y # z ) ) )
52	51	rgen3 2595	. . 3 |- A. x e. CC A. y e. CC A. z e. CC ( x # y -> ( x # z \/ y # z ) )
53		apti 8730	. . . . 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 2594	. . 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 7398	. 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 1182	1 |- # TAp CC

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   C_ wss 3174   <.cop 3646   class class class wbr 4059   {copab 4120   X. cxp 4691   Rel wrel 4698   ` cfv 5290  (class class class)co 5967   1stc1st 6247   2ndc2nd 6248   TAp wtap 7396   CCcc 7958   RRcr 7959   _ici 7962   + caddc 7963   x. cmul 7965   #_ℝ creap 8682   # cap 8689

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pap 7395  df-tap 7397  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690

This theorem is referenced by: (None)