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Theorem xqltnle 10336

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 10313	. . . 4 $/\$
2	1	adantll 476	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
3		simplr 528	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
4		qre 9690	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
6	5	ltpnfd 9847	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
7		simpr 110	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
8	6, 7	breqtrrd 4057	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
9	5	renepnfd 8070	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
10	9	neneqd 2385	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
11	5	rexrd 8069	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
12		xgepnf 9882	. . . . . . 7
13	11, 12	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
14	10, 13	mtbird 674	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
15	7, 14	eqnbrtrd 4047	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
16	8, 15	2thd 175	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
17		simplr 528	. . 3 $\/$ $/\$ $\/$ $/\$ $\/$
18	2, 16, 17	mpjaodan 799	. 2 $\/$ $/\$ $\/$ $/\$
19		simpr 110	. . . 4 $\/$ $/\$ $\/$ $/\$
20		qre 9690	. . . . . . . . 9
21	20	rexrd 8069	. . . . . . . 8
22		pnfxr 8072	. . . . . . . . 9
23		eleq1 2256	. . . . . . . . 9
24	22, 23	mpbiri 168	. . . . . . . 8
25	21, 24	jaoi 717	. . . . . . 7 $\/$
26	25	adantl 277	. . . . . 6 $\/$ $/\$ $\/$
27		pnfnlt 9853	. . . . . 6
28	26, 27	syl 14	. . . . 5 $\/$ $/\$ $\/$
29	28	adantr 276	. . . 4 $\/$ $/\$ $\/$ $/\$
30	19, 29	eqnbrtrd 4047	. . 3 $\/$ $/\$ $\/$ $/\$
31		pnfge 9855	. . . . . . 7
32	26, 31	syl 14	. . . . . 6 $\/$ $/\$ $\/$
33	32	adantr 276	. . . . 5 $\/$ $/\$ $\/$ $/\$
34	33, 19	breqtrrd 4057	. . . 4 $\/$ $/\$ $\/$ $/\$
35	34	notnotd 631	. . 3 $\/$ $/\$ $\/$ $/\$
36	30, 35	2falsed 703	. 2 $\/$ $/\$ $\/$ $/\$
37		simpl 109	. 2 $\/$ $/\$ $\/$ $\/$
38	18, 36, 37	mpjaodan 799	1 $\/$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164 class class class wbr 4029

cr 7871

cpnf 8051

cxr 8053

clt 8054

cle 8055

cq 9684

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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990

This theorem depends on definitions: df-bi 117 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-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-q 9685 df-rp 9720

This theorem is referenced by: pcadd2 12479

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