cndcap - Mathbox for Jim Kingdon

	Mathbox for Jim Kingdon		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > Mathboxes > cndcap		Unicode version

Theorem cndcap 17014

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 4129	. . . . . . 7 # #
2	1	dcbid 850	. . . . . 6 DECID # DECID #
3		breq1 4128	. . . . . . . . 9 # #
4	3	dcbid 850	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2550	. . . . . . 7 DECID # DECID #
6		triap 16983	. . . . . . . . . . 11 $/\$ $\/$ $\/$ DECID #
7	6	ralbidva 2546	. . . . . . . . . 10 $\/$ $\/$ DECID #
8	7	ralbiia 2564	. . . . . . . . 9 $\/$ $\/$ DECID #
9	8	biimpi 120	. . . . . . . 8 $\/$ $\/$ DECID #
10	9	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
11		simprl 535	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
12	11	recld 11682	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2934	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 537	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11682	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2934	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4129	. . . . . . 7 # #
18	17	dcbid 850	. . . . . 6 DECID # DECID #
19		breq1 4128	. . . . . . . . 9 # #
20	19	dcbid 850	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2550	. . . . . . 7 DECID # DECID #
22	11	imcld 11683	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2934	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11683	. . . . . 6 $\/$ $\/$ $/\$ $/\$
25	18, 23, 24	rspcdva 2934	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
26		dcor 948	. . . . 5 DECID # DECID # DECID # $\/$ #
27	16, 25, 26	sylc 62	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # $\/$ #
28		cnreim 11722	. . . . . 6 $/\$ # # $\/$ #
29	28	dcbid 850	. . . . 5 $/\$ DECID # DECID # $\/$ #
30	29	adantl 277	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID # DECID # $\/$ #
31	27, 30	mpbird 167	. . 3 $\/$ $\/$ $/\$ $/\$ DECID #
32	31	ralrimivva 2632	. 2 $\/$ $\/$ DECID #
33		breq2 4129	. . . . . 6 # #
34	33	dcbid 850	. . . . 5 DECID # DECID #
35		breq1 4128	. . . . . . . 8 # #
36	35	dcbid 850	. . . . . . 7 DECID # DECID #
37	36	ralbidv 2550	. . . . . 6 DECID # DECID #
38		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
39		simprl 535	. . . . . . 7 DECID # $/\$ $/\$
40	39	recnd 8344	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2934	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 537	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8344	. . . . 5 DECID # $/\$ $/\$
44	34, 41, 43	rspcdva 2934	. . . 4 DECID # $/\$ $/\$ DECID #
45	6	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
46	44, 45	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
47	46	ralrimivva 2632	. 2 DECID # $\/$ $\/$
48	32, 47	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints: $/\$ wa 104

wb 105 $\/$ wo 720 DECID wdc 846 $\/$ w3o 1008

wceq 1402

wcel 2209

wral 2528 class class class wbr 4125

cfv 5372

cc 8167

cr 8168

clt 8350 # cap 8899

cre 11583

cim 11584

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-2 9342 df-cj 11585 df-re 11586 df-im 11587

This theorem is referenced by: (None)

W3C validator