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 17109

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 4134	. . . . . . 7 # #
2	1	dcbid 850	. . . . . 6 DECID # DECID #
3		breq1 4133	. . . . . . . . 9 # #
4	3	dcbid 850	. . . . . . . 8 DECID # DECID #
5	4	ralbidv 2550	. . . . . . 7 DECID # DECID #
6		triap 17078	. . . . . . . . . . 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 11704	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
13	5, 10, 12	rspcdva 2934	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
14		simprr 537	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
15	14	recld 11704	. . . . . 6 $\/$ $\/$ $/\$ $/\$
16	2, 13, 15	rspcdva 2934	. . . . 5 $\/$ $\/$ $/\$ $/\$ DECID #
17		breq2 4134	. . . . . . 7 # #
18	17	dcbid 850	. . . . . 6 DECID # DECID #
19		breq1 4133	. . . . . . . . 9 # #
20	19	dcbid 850	. . . . . . . 8 DECID # DECID #
21	20	ralbidv 2550	. . . . . . 7 DECID # DECID #
22	11	imcld 11705	. . . . . . 7 $\/$ $\/$ $/\$ $/\$
23	21, 10, 22	rspcdva 2934	. . . . . 6 $\/$ $\/$ $/\$ $/\$ DECID #
24	14	imcld 11705	. . . . . 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 11744	. . . . . 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 4134	. . . . . 6 # #
34	33	dcbid 850	. . . . 5 DECID # DECID #
35		breq1 4133	. . . . . . . 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 8354	. . . . . 6 DECID # $/\$ $/\$
41	37, 38, 40	rspcdva 2934	. . . . 5 DECID # $/\$ $/\$ DECID #
42		simprr 537	. . . . . 6 DECID # $/\$ $/\$
43	42	recnd 8354	. . . . 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

This proof depends on syntax axioms: $/\$ wa 104

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

wceq 1402

wcel 2209

wral 2528 class class class wbr 4130

cfv 5377

cc 8177

cr 8178

clt 8360 # cap 8909

cre 11605

cim 11606

This proof depends on 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 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof 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 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-po 4441 df-iso 4442 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-2 9363 df-cj 11607 df-re 11608 df-im 11609

This theorem is used by: (None)

W3C validator