cndcap - Mathbox for Jim Kingdon

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Theorem cndcap 15104

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 4019	. . . . . . 7 # #
2	1	dcbid 839	. . . . . 6 DECID # DECID #
3		breq1 4018	. . . . . . . . 9 # #
4	3	dcbid 839	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2487	. . . . . . 7 DECID # DECID #
6		triap 15074	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2483	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2501	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 529	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 10961	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2858	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 531	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 10961	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2858	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4019	. . . . . . 7 # #
18	17	dcbid 839	. . . . . 6 DECID # DECID #
19		breq1 4018	. . . . . . . . 9 # #
20	19	dcbid 839	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2487	. . . . . . 7 DECID # DECID #
22	11	imcld 10962	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2858	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 10962	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2858	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 936	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11001	. . . . . 6 $/\$ # # $\/$ #
29	28	dcbid 839	. . . . 5 $/\$ DECID # DECID # $\/$ #
30	29	adantl 277	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # DECID # $\/$ #
31	27, 30	mpbird 167	. . 3 $\/$ $\/$ $/\$ $/\$ DECID #
32	31	ralrimivva 2569	. 2 $\/$ $\/$ DECID #
33		breq2 4019	. . . . . 6 # #
34	33	dcbid 839	. . . . 5 DECID # DECID #
35		breq1 4018	. . . . . . . 8 # #
36	35	dcbid 839	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2487	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 529	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8000	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2858	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 531	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8000	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2858	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2569	. 2 DECID # $\/$ $\/$
48	32, 47	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 709 DECID wdc 835 $\/$ w3o 978

wceq 1363

wcel 2158

wral 2465 class class class wbr 4015

cfv 5228

cc 7823

cr 7824

clt 8006 # cap 8552

cre 10863

cim 10864

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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-mulrcl 7924 ax-addcom 7925 ax-mulcom 7926 ax-addass 7927 ax-mulass 7928 ax-distr 7929 ax-i2m1 7930 ax-0lt1 7931 ax-1rid 7932 ax-0id 7933 ax-rnegex 7934 ax-precex 7935 ax-cnre 7936 ax-pre-ltirr 7937 ax-pre-ltwlin 7938 ax-pre-lttrn 7939 ax-pre-apti 7940 ax-pre-ltadd 7941 ax-pre-mulgt0 7942 ax-pre-mulext 7943

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-xr 8010 df-ltxr 8011 df-le 8012 df-sub 8144 df-neg 8145 df-reap 8546 df-ap 8553 df-div 8644 df-2 8992 df-cj 10865 df-re 10866 df-im 10867

This theorem is referenced by: (None)

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