cndcap - Mathbox for Jim Kingdon

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Theorem cndcap 16970

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 4118	. . . . . . 7 # #
2	1	dcbid 846	. . . . . 6 DECID # DECID #
3		breq1 4117	. . . . . . . . 9 # #
4	3	dcbid 846	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2544	. . . . . . 7 DECID # DECID #
6		triap 16939	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2540	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2558	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 531	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 11648	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2928	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 533	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11648	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2928	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4118	. . . . . . 7 # #
18	17	dcbid 846	. . . . . 6 DECID # DECID #
19		breq1 4117	. . . . . . . . 9 # #
20	19	dcbid 846	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2544	. . . . . . 7 DECID # DECID #
22	11	imcld 11649	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2928	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11649	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2928	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 944	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11688	. . . . . 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 2626	. 2 $\/$ $\/$ DECID #
33		breq2 4118	. . . . . 6 # #
34	33	dcbid 846	. . . . 5 DECID # DECID #
35		breq1 4117	. . . . . . . 8 # #
36	35	dcbid 846	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2544	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 531	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8318	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2928	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 533	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8318	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2928	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2626	. 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 2205

wral 2522 class class class wbr 4114

cfv 5357

cc 8141

cr 8142

clt 8324 # cap 8872

cre 11550

cim 11551

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

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 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-2 9313 df-cj 11552 df-re 11553 df-im 11554

This theorem is referenced by: trimul0or 16971

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