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Theorem xqltnle 10651

Description: "Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or

. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in NN0* or

, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)

**Assertion**
Ref	Expression
xqltnle	$\/$ $/\$ $\/$

**Proof of Theorem xqltnle**
Step	Hyp	Ref	Expression
1		qltnle 10627	. . . 4 $/\$
2	1	adantll 476	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
3		simplr 529	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
4		qre 9975	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
6	5	ltpnfd 10133	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
7		simpr 110	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
8	6, 7	breqtrrd 4142	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
9	5	renepnfd 8340	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
10	9	neneqd 2435	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
11	5	rexrd 8339	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
12		xgepnf 10168	. . . . . . 7
13	11, 12	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
14	10, 13	mtbird 680	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
15	7, 14	eqnbrtrd 4132	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
16	8, 15	2thd 175	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
17		simplr 529	. . 3 $\/$ $/\$ $\/$ $/\$ $\/$
18	2, 16, 17	mpjaodan 806	. 2 $\/$ $/\$ $\/$ $/\$
19		simpr 110	. . . 4 $\/$ $/\$ $\/$ $/\$
20		qre 9975	. . . . . . . . 9
21	20	rexrd 8339	. . . . . . . 8
22		pnfxr 8342	. . . . . . . . 9
23		eleq1 2297	. . . . . . . . 9
24	22, 23	mpbiri 168	. . . . . . . 8
25	21, 24	jaoi 724	. . . . . . 7 $\/$
26	25	adantl 277	. . . . . 6 $\/$ $/\$ $\/$
27		pnfnlt 10139	. . . . . 6
28	26, 27	syl 14	. . . . 5 $\/$ $/\$ $\/$
29	28	adantr 276	. . . 4 $\/$ $/\$ $\/$ $/\$
30	19, 29	eqnbrtrd 4132	. . 3 $\/$ $/\$ $\/$ $/\$
31		pnfge 10141	. . . . . . 7
32	26, 31	syl 14	. . . . . 6 $\/$ $/\$ $\/$
33	32	adantr 276	. . . . 5 $\/$ $/\$ $\/$ $/\$
34	33, 19	breqtrrd 4142	. . . 4 $\/$ $/\$ $\/$ $/\$
35	34	notnotd 635	. . 3 $\/$ $/\$ $\/$ $/\$
36	30, 35	2falsed 710	. 2 $\/$ $/\$ $\/$ $/\$
37		simpl 109	. 2 $\/$ $/\$ $\/$ $\/$
38	18, 36, 37	mpjaodan 806	1 $\/$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2205 class class class wbr 4114

cr 8142

cpnf 8321

cxr 8323

clt 8324

cle 8325

cq 9969

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-n0 9514 df-z 9595 df-q 9970 df-rp 10005

This theorem is referenced by: pcadd2 13064

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