reap0 - Mathbox for Jim Kingdon

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Theorem reap0 13592

Description: Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.)

**Assertion**
Ref	Expression
reap0	$\/$ $\/$ DECID #

Distinct variable group:

**Proof of Theorem reap0**
Step	Hyp	Ref	Expression
1		simpl 108	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
2		simpr 109	. . . . . 6 $\/$ $\/$ $/\$
3		0re 7861	. . . . . 6
4		breq1 3968	. . . . . . . 8
5		equequ1 1692	. . . . . . . 8
6		breq2 3969	. . . . . . . 8
7	4, 5, 6	3orbi123d 1293	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 3969	. . . . . . . 8
9		eqeq2 2167	. . . . . . . 8
10		breq1 3968	. . . . . . . 8
11	8, 9, 10	3orbi123d 1293	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2829	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 410	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 13563	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 410	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 146	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2530	. 2 $\/$ $\/$ DECID #
19		breq1 3968	. . . . . . 7 # #
20	19	dcbid 824	. . . . . 6 DECID # DECID #
21		simpl 108	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8122	. . . . . . 7 $/\$
23	22	adantl 275	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2821	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 521	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 7889	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 522	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 7889	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8501	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 409	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 824	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 146	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 13563	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 275	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 166	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2539	. 2 DECID # $\/$ $\/$
37	18, 36	impbii 125	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 DECID wdc 820 $\/$ w3o 962

wceq 1335

wcel 2128

wral 2435 class class class wbr 3965 (class class class)co 5818

cc 7713

cr 7714

cc0 7715

clt 7895

cmin 8029 # cap 8439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-cnex 7806 ax-resscn 7807 ax-1cn 7808 ax-1re 7809 ax-icn 7810 ax-addcl 7811 ax-addrcl 7812 ax-mulcl 7813 ax-mulrcl 7814 ax-addcom 7815 ax-mulcom 7816 ax-addass 7817 ax-mulass 7818 ax-distr 7819 ax-i2m1 7820 ax-0lt1 7821 ax-1rid 7822 ax-0id 7823 ax-rnegex 7824 ax-precex 7825 ax-cnre 7826 ax-pre-ltirr 7827 ax-pre-lttrn 7829 ax-pre-apti 7830 ax-pre-ltadd 7831 ax-pre-mulgt0 7832

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4252 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-iota 5132 df-fun 5169 df-fv 5175 df-riota 5774 df-ov 5821 df-oprab 5822 df-mpo 5823 df-pnf 7897 df-mnf 7898 df-ltxr 7900 df-sub 8031 df-neg 8032 df-reap 8433 df-ap 8440

This theorem is referenced by: dcapnconstALT 13595

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