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Theorem xqltnle 10487

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 10463	. . . 4 $/\$
2	1	adantll 476	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
3		simplr 528	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
4		qre 9820	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
6	5	ltpnfd 9977	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
7		simpr 110	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
8	6, 7	breqtrrd 4111	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
9	5	renepnfd 8197	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
10	9	neneqd 2421	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
11	5	rexrd 8196	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
12		xgepnf 10012	. . . . . . 7
13	11, 12	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
14	10, 13	mtbird 677	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
15	7, 14	eqnbrtrd 4101	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
16	8, 15	2thd 175	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
17		simplr 528	. . 3 $\/$ $/\$ $\/$ $/\$ $\/$
18	2, 16, 17	mpjaodan 803	. 2 $\/$ $/\$ $\/$ $/\$
19		simpr 110	. . . 4 $\/$ $/\$ $\/$ $/\$
20		qre 9820	. . . . . . . . 9
21	20	rexrd 8196	. . . . . . . 8
22		pnfxr 8199	. . . . . . . . 9
23		eleq1 2292	. . . . . . . . 9
24	22, 23	mpbiri 168	. . . . . . . 8
25	21, 24	jaoi 721	. . . . . . 7 $\/$
26	25	adantl 277	. . . . . 6 $\/$ $/\$ $\/$
27		pnfnlt 9983	. . . . . 6
28	26, 27	syl 14	. . . . 5 $\/$ $/\$ $\/$
29	28	adantr 276	. . . 4 $\/$ $/\$ $\/$ $/\$
30	19, 29	eqnbrtrd 4101	. . 3 $\/$ $/\$ $\/$ $/\$
31		pnfge 9985	. . . . . . 7
32	26, 31	syl 14	. . . . . 6 $\/$ $/\$ $\/$
33	32	adantr 276	. . . . 5 $\/$ $/\$ $\/$ $/\$
34	33, 19	breqtrrd 4111	. . . 4 $\/$ $/\$ $\/$ $/\$
35	34	notnotd 633	. . 3 $\/$ $/\$ $\/$ $/\$
36	30, 35	2falsed 707	. 2 $\/$ $/\$ $\/$ $/\$
37		simpl 109	. 2 $\/$ $/\$ $\/$ $\/$
38	18, 36, 37	mpjaodan 803	1 $\/$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 713

wceq 1395

wcel 2200 class class class wbr 4083

cr 7998

cpnf 8178

cxr 8180

clt 8181

cle 8182

cq 9814

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-n0 9370 df-z 9447 df-q 9815 df-rp 9850

This theorem is referenced by: pcadd2 12864

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