reap0 - Mathbox for Jim Kingdon

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Theorem reap0 15618

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 8021	. . . . . 6
4		breq1 4033	. . . . . . . 8
5		equequ1 1723	. . . . . . . 8
6		breq2 4034	. . . . . . . 8
7	4, 5, 6	3orbi123d 1322	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 4034	. . . . . . . 8
9		eqeq2 2203	. . . . . . . 8
10		breq1 4033	. . . . . . . 8
11	8, 9, 10	3orbi123d 1322	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2878	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 413	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 15589	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 413	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 147	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2567	. 2 $\/$ $\/$ DECID #
19		breq1 4033	. . . . . . 7 # #
20	19	dcbid 839	. . . . . 6 DECID # DECID #
21		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8285	. . . . . . 7 $/\$
23	22	adantl 277	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2870	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 529	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 8050	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 531	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 8050	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8664	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 411	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 839	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 147	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 15589	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2576	. 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 835 $\/$ w3o 979

wceq 1364

wcel 2164

wral 2472 class class class wbr 4030 (class class class)co 5919

cc 7872

cr 7873

cc0 7874

clt 8056

cmin 8192 # cap 8602

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-iota 5216 df-fun 5257 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-pnf 8058 df-mnf 8059 df-ltxr 8061 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603

This theorem is referenced by: dcapnconstALT 15622

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