reap0 - Mathbox for Jim Kingdon

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Theorem reap0 17013

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 8316	. . . . . 6
4		breq1 4128	. . . . . . . 8
5		equequ1 1764	. . . . . . . 8
6		breq2 4129	. . . . . . . 8
7	4, 5, 6	3orbi123d 1352	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		breq2 4129	. . . . . . . 8
9		eqeq2 2248	. . . . . . . 8
10		breq1 4128	. . . . . . . 8
11	8, 9, 10	3orbi123d 1352	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
12	7, 11	rspc2v 2943	. . . . . 6 $/\$ $\/$ $\/$ $\/$ $\/$
13	2, 3, 12	sylancl 417	. . . . 5 $\/$ $\/$ $/\$ $\/$ $\/$ $\/$ $\/$
14	1, 13	mpd 13	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$
15		triap 16983	. . . . 5 $/\$ $\/$ $\/$ DECID #
16	2, 3, 15	sylancl 417	. . . 4 $\/$ $\/$ $/\$ $\/$ $\/$ DECID #
17	14, 16	mpbid 147	. . 3 $\/$ $\/$ $/\$ DECID #
18	17	ralrimiva 2623	. 2 $\/$ $\/$ DECID #
19		breq1 4128	. . . . . . 7 # #
20	19	dcbid 850	. . . . . 6 DECID # DECID #
21		simpl 109	. . . . . 6 DECID # $/\$ $/\$ DECID #
22		resubcl 8580	. . . . . . 7 $/\$
23	22	adantl 277	. . . . . 6 DECID # $/\$ $/\$
24	20, 21, 23	rspcdva 2934	. . . . 5 DECID # $/\$ $/\$ DECID #
25		simprl 535	. . . . . . . 8 DECID # $/\$ $/\$
26	25	recnd 8344	. . . . . . 7 DECID # $/\$ $/\$
27		simprr 537	. . . . . . . 8 DECID # $/\$ $/\$
28	27	recnd 8344	. . . . . . 7 DECID # $/\$ $/\$
29		subap0 8961	. . . . . . 7 $/\$ # #
30	26, 28, 29	syl2anc 415	. . . . . 6 DECID # $/\$ $/\$ # #
31	30	dcbid 850	. . . . 5 DECID # $/\$ $/\$ DECID # DECID #
32	24, 31	mpbid 147	. . . 4 DECID # $/\$ $/\$ DECID #
33		triap 16983	. . . . 5 $/\$ $\/$ $\/$ DECID #
34	33	adantl 277	. . . 4 DECID # $/\$ $/\$ $\/$ $\/$ DECID #
35	32, 34	mpbird 167	. . 3 DECID # $/\$ $/\$ $\/$ $\/$
36	35	ralrimivva 2632	. 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 846 $\/$ w3o 1008

wceq 1402

wcel 2209

wral 2528 class class class wbr 4125 (class class class)co 6075

cc 8167

cr 8168

cc0 8169

clt 8350

cmin 8487 # cap 8899

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 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 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286

This theorem 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-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-ltxr 8355 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900

This theorem is referenced by: dcapnconstALT 17017

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