cndcap - Mathbox for Jim Kingdon

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Theorem cndcap 16599

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 4090	. . . . . . 7 # #
2	1	dcbid 843	. . . . . 6 DECID # DECID #
3		breq1 4089	. . . . . . . . 9 # #
4	3	dcbid 843	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2530	. . . . . . 7 DECID # DECID #
6		triap 16569	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2526	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2544	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 529	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 11489	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2913	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 531	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11489	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2913	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4090	. . . . . . 7 # #
18	17	dcbid 843	. . . . . 6 DECID # DECID #
19		breq1 4089	. . . . . . . . 9 # #
20	19	dcbid 843	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2530	. . . . . . 7 DECID # DECID #
22	11	imcld 11490	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2913	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11490	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2913	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 941	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11529	. . . . . 6 $/\$ # # $\/$ #
29	28	dcbid 843	. . . . 5 $/\$ DECID # DECID # $\/$ #
30	29	adantl 277	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # DECID # $\/$ #
31	27, 30	mpbird 167	. . 3 $\/$ $\/$ $/\$ $/\$ DECID #
32	31	ralrimivva 2612	. 2 $\/$ $\/$ DECID #
33		breq2 4090	. . . . . 6 # #
34	33	dcbid 843	. . . . 5 DECID # DECID #
35		breq1 4089	. . . . . . . 8 # #
36	35	dcbid 843	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2530	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 529	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8198	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2913	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 531	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8198	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2913	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2612	. 2 DECID # $\/$ $\/$
48	32, 47	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 713 DECID wdc 839 $\/$ w3o 1001

wceq 1395

wcel 2200

wral 2508 class class class wbr 4086

cfv 5324

cc 8020

cr 8021

clt 8204 # cap 8751

cre 11391

cim 11392

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-po 4391 df-iso 4392 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-2 9192 df-cj 11393 df-re 11394 df-im 11395

This theorem is referenced by: (None)

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