cndcap - Mathbox for Jim Kingdon

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Theorem cndcap 16035

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 4049	. . . . . . 7 # #
2	1	dcbid 840	. . . . . 6 DECID # DECID #
3		breq1 4048	. . . . . . . . 9 # #
4	3	dcbid 840	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2506	. . . . . . 7 DECID # DECID #
6		triap 16005	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2502	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2520	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 529	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 11282	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2882	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 531	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11282	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2882	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4049	. . . . . . 7 # #
18	17	dcbid 840	. . . . . 6 DECID # DECID #
19		breq1 4048	. . . . . . . . 9 # #
20	19	dcbid 840	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2506	. . . . . . 7 DECID # DECID #
22	11	imcld 11283	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2882	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11283	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2882	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 938	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11322	. . . . . 6 $/\$ # # $\/$ #
29	28	dcbid 840	. . . . 5 $/\$ DECID # DECID # $\/$ #
30	29	adantl 277	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # DECID # $\/$ #
31	27, 30	mpbird 167	. . 3 $\/$ $\/$ $/\$ $/\$ DECID #
32	31	ralrimivva 2588	. 2 $\/$ $\/$ DECID #
33		breq2 4049	. . . . . 6 # #
34	33	dcbid 840	. . . . 5 DECID # DECID #
35		breq1 4048	. . . . . . . 8 # #
36	35	dcbid 840	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2506	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 529	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8103	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2882	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 531	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8103	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2882	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2588	. 2 DECID # $\/$ $\/$
48	32, 47	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 710 DECID wdc 836 $\/$ w3o 980

wceq 1373

wcel 2176

wral 2484 class class class wbr 4045

cfv 5272

cc 7925

cr 7926

clt 8109 # cap 8656

cre 11184

cim 11185

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4046 df-opab 4107 df-mpt 4108 df-id 4341 df-po 4344 df-iso 4345 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-2 9097 df-cj 11186 df-re 11187 df-im 11188

This theorem is referenced by: (None)

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