cndcap - Mathbox for Jim Kingdon

	Mathbox for Jim Kingdon		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > Mathboxes > cndcap		Unicode version

Theorem cndcap 16792

Description: Real number trichotomy is equivalent to decidability of complex number apartness. (Contributed by Jim Kingdon, 10-Apr-2025.)

**Assertion**
Ref	Expression
cndcap	$\/$ $\/$ DECID #

Distinct variable group:

**Proof of Theorem cndcap**
Step	Hyp	Ref	Expression
1		breq2 4097	. . . . . . 7 # #
2	1	dcbid 846	. . . . . 6 DECID # DECID #
3		breq1 4096	. . . . . . . . 9 # #
4	3	dcbid 846	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2533	. . . . . . 7 DECID # DECID #
6		triap 16761	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2529	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2547	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 531	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 11578	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2916	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 533	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11578	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2916	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4097	. . . . . . 7 # #
18	17	dcbid 846	. . . . . 6 DECID # DECID #
19		breq1 4096	. . . . . . . . 9 # #
20	19	dcbid 846	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2533	. . . . . . 7 DECID # DECID #
22	11	imcld 11579	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2916	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11579	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2916	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 944	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11618	. . . . . 6 $/\$ # # $\/$ #
29	28	dcbid 846	. . . . 5 $/\$ DECID # DECID # $\/$ #
30	29	adantl 277	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # DECID # $\/$ #
31	27, 30	mpbird 167	. . 3 $\/$ $\/$ $/\$ $/\$ DECID #
32	31	ralrimivva 2615	. 2 $\/$ $\/$ DECID #
33		breq2 4097	. . . . . 6 # #
34	33	dcbid 846	. . . . 5 DECID # DECID #
35		breq1 4096	. . . . . . . 8 # #
36	35	dcbid 846	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2533	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 531	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8267	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2916	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 533	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8267	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2916	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2615	. 2 DECID # $\/$ $\/$
48	32, 47	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 716 DECID wdc 842 $\/$ w3o 1004

wceq 1398

wcel 2202

wral 2511 class class class wbr 4093

cfv 5333

cc 8090

cr 8091

clt 8273 # cap 8820

cre 11480

cim 11481

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-precex 8202 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-apti 8207 ax-pre-ltadd 8208 ax-pre-mulgt0 8209 ax-pre-mulext 8210

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-po 4399 df-iso 4400 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-reap 8814 df-ap 8821 df-div 8912 df-2 9261 df-cj 11482 df-re 11483 df-im 11484

This theorem is referenced by: (None)

W3C validator