cndcap - Mathbox for Jim Kingdon

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Theorem cndcap 14892

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 4009	. . . . . . 7 # #
2	1	dcbid 838	. . . . . 6 DECID # DECID #
3		breq1 4008	. . . . . . . . 9 # #
4	3	dcbid 838	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2477	. . . . . . 7 DECID # DECID #
6		triap 14862	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2473	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2491	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 529	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 10949	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2848	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 531	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 10949	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2848	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4009	. . . . . . 7 # #
18	17	dcbid 838	. . . . . 6 DECID # DECID #
19		breq1 4008	. . . . . . . . 9 # #
20	19	dcbid 838	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2477	. . . . . . 7 DECID # DECID #
22	11	imcld 10950	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2848	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 10950	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2848	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 935	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 10989	. . . . . 6 $/\$ # # $\/$ #
29	28	dcbid 838	. . . . 5 $/\$ DECID # DECID # $\/$ #
30	29	adantl 277	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # DECID # $\/$ #
31	27, 30	mpbird 167	. . 3 $\/$ $\/$ $/\$ $/\$ DECID #
32	31	ralrimivva 2559	. 2 $\/$ $\/$ DECID #
33		breq2 4009	. . . . . 6 # #
34	33	dcbid 838	. . . . 5 DECID # DECID #
35		breq1 4008	. . . . . . . 8 # #
36	35	dcbid 838	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2477	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 529	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 7988	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2848	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 531	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 7988	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2848	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2559	. 2 DECID # $\/$ $\/$
48	32, 47	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 708 DECID wdc 834 $\/$ w3o 977

wceq 1353

wcel 2148

wral 2455 class class class wbr 4005

cfv 5218

cc 7811

cr 7812

clt 7994 # cap 8540

cre 10851

cim 10852

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-2 8980 df-cj 10853 df-re 10854 df-im 10855

This theorem is referenced by: (None)

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