reap0 - Mathbox for Jim Kingdon

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Theorem reap0 16199

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 109	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$
2		simpr 110	. . . . . 6 $\/$ $\/$ $/\$
3		0re 8107	. . . . . 6
4		breq1 4062	. . . . . . . 8
5		equequ1 1736	. . . . . . . 8
6		breq2 4063	. . . . . . . 8
7	4, 5, 6	3orbi123d 1324	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 4063	. . . . . . . 8
9		eqeq2 2217	. . . . . . . 8
10		breq1 4062	. . . . . . . 8
11	8, 9, 10	3orbi123d 1324	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2897	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 413	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 16170	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 413	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 147	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2581	. 2 $\/$ $\/$ DECID #
19		breq1 4062	. . . . . . 7 # #
20	19	dcbid 840	. . . . . 6 DECID # DECID #
21		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8371	. . . . . . 7 $/\$
23	22	adantl 277	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2889	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 529	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 8136	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 531	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 8136	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8751	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 411	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 840	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 147	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 16170	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2590	. 2 DECID # $\/$ $\/$
37	18, 36	impbii 126	1 $\/$ $\/$ DECID #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 DECID wdc 836 $\/$ w3o 980

wceq 1373

wcel 2178

wral 2486 class class class wbr 4059 (class class class)co 5967

cc 7958

cr 7959

cc0 7960

clt 8142

cmin 8278 # cap 8689

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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077

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 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-id 4358 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-iota 5251 df-fun 5292 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pnf 8144 df-mnf 8145 df-ltxr 8147 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690

This theorem is referenced by: dcapnconstALT 16203

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