reap0 - Mathbox for Jim Kingdon

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Theorem reap0 14090

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 7920	. . . . . 6
4		breq1 3992	. . . . . . . 8
5		equequ1 1705	. . . . . . . 8
6		breq2 3993	. . . . . . . 8
7	4, 5, 6	3orbi123d 1306	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 3993	. . . . . . . 8
9		eqeq2 2180	. . . . . . . 8
10		breq1 3992	. . . . . . . 8
11	8, 9, 10	3orbi123d 1306	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2847	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 411	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 14061	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 411	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 146	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2543	. 2 $\/$ $\/$ DECID #
19		breq1 3992	. . . . . . 7 # #
20	19	dcbid 833	. . . . . 6 DECID # DECID #
21		simpl 108	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8183	. . . . . . 7 $/\$
23	22	adantl 275	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2839	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 526	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 7948	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 527	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 7948	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8562	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 409	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 833	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 146	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 14061	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 275	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 166	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2552	. 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 829 $\/$ w3o 972

wceq 1348

wcel 2141

wral 2448 class class class wbr 3989 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

clt 7954

cmin 8090 # cap 8500

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501

This theorem is referenced by: dcapnconstALT 14093

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