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Theorem xqltnle 10702

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 10678	. . . 4 $/\$
2	1	adantll 480	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
3		simplr 533	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
4		qre 10025	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
6	5	ltpnfd 10183	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
7		simpr 110	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
8	6, 7	breqtrrd 4158	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
9	5	renepnfd 8376	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
10	9	neneqd 2441	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
11	5	rexrd 8375	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
12		xgepnf 10218	. . . . . . 7
13	11, 12	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
14	10, 13	mtbird 684	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
15	7, 14	eqnbrtrd 4148	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
16	8, 15	2thd 175	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
17		simplr 533	. . 3 $\/$ $/\$ $\/$ $/\$ $\/$
18	2, 16, 17	mpjaodan 810	. 2 $\/$ $/\$ $\/$ $/\$
19		simpr 110	. . . 4 $\/$ $/\$ $\/$ $/\$
20		qre 10025	. . . . . . . . 9
21	20	rexrd 8375	. . . . . . . 8
22		pnfxr 8378	. . . . . . . . 9
23		eleq1 2301	. . . . . . . . 9
24	22, 23	mpbiri 168	. . . . . . . 8
25	21, 24	jaoi 728	. . . . . . 7 $\/$
26	25	adantl 277	. . . . . 6 $\/$ $/\$ $\/$
27		pnfnlt 10189	. . . . . 6
28	26, 27	syl 14	. . . . 5 $\/$ $/\$ $\/$
29	28	adantr 276	. . . 4 $\/$ $/\$ $\/$ $/\$
30	19, 29	eqnbrtrd 4148	. . 3 $\/$ $/\$ $\/$ $/\$
31		pnfge 10191	. . . . . . 7
32	26, 31	syl 14	. . . . . 6 $\/$ $/\$ $\/$
33	32	adantr 276	. . . . 5 $\/$ $/\$ $\/$ $/\$
34	33, 19	breqtrrd 4158	. . . 4 $\/$ $/\$ $\/$ $/\$
35	34	notnotd 639	. . 3 $\/$ $/\$ $\/$ $/\$
36	30, 35	2falsed 714	. 2 $\/$ $/\$ $\/$ $/\$
37		simpl 109	. 2 $\/$ $/\$ $\/$ $\/$
38	18, 36, 37	mpjaodan 810	1 $\/$ $/\$ $\/$

Colors of variables: wff set class

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209 class class class wbr 4130

cr 8178

cpnf 8357

cxr 8359

clt 8360

cle 8361

cq 10019

This proof depends on 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 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof depends on definitions: df-bi 117 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-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-po 4441 df-iso 4442 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-n0 9564 df-z 9645 df-q 10020 df-rp 10055

This theorem is used by: pcadd2 13120

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