cndcap - Mathbox for Jim Kingdon

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Theorem cndcap 15790

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 4038	. . . . . . 7 # #
2	1	dcbid 839	. . . . . 6 DECID # DECID #
3		breq1 4037	. . . . . . . . 9 # #
4	3	dcbid 839	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2497	. . . . . . 7 DECID # DECID #
6		triap 15760	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2493	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2511	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 529	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 11120	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2873	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 531	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11120	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2873	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4038	. . . . . . 7 # #
18	17	dcbid 839	. . . . . 6 DECID # DECID #
19		breq1 4037	. . . . . . . . 9 # #
20	19	dcbid 839	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2497	. . . . . . 7 DECID # DECID #
22	11	imcld 11121	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2873	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11121	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2873	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 937	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11160	. . . . . 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 2579	. 2 $\/$ $\/$ DECID #
33		breq2 4038	. . . . . 6 # #
34	33	dcbid 839	. . . . 5 DECID # DECID #
35		breq1 4037	. . . . . . . 8 # #
36	35	dcbid 839	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2497	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 529	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8072	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2873	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 531	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8072	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2873	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2579	. 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 979

wceq 1364

wcel 2167

wral 2475 class class class wbr 4034

cfv 5259

cc 7894

cr 7895

clt 8078 # cap 8625

cre 11022

cim 11023

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-2 9066 df-cj 11024 df-re 11025 df-im 11026

This theorem is referenced by: (None)

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