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 ℕ₀^* or ℝ^*, 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	⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Proof of Theorem xqltnle

Step	Hyp	Ref	Expression
1		qltnle 10313	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
2	1	adantll 476	. . 3 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
3		simplr 528	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℚ)
4		qre 9690	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
5	3, 4	syl 14	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ)
6	5	ltpnfd 9847	. . . . 5 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 < +∞)
7		simpr 110	. . . . 5 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐵 = +∞)
8	6, 7	breqtrrd 4057	. . . 4 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
9	5	renepnfd 8070	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ≠ +∞)
10	9	neneqd 2385	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ 𝐴 = +∞)
11	5	rexrd 8069	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*)
12		xgepnf 9882	. . . . . . 7 ⊢ (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))
13	11, 12	syl 14	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))
14	10, 13	mtbird 674	. . . . 5 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ +∞ ≤ 𝐴)
15	7, 14	eqnbrtrd 4047	. . . 4 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ 𝐵 ≤ 𝐴)
16	8, 15	2thd 175	. . 3 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
17		simplr 528	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) → (𝐵 ∈ ℚ ∨ 𝐵 = +∞))
18	2, 16, 17	mpjaodan 799	. 2 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
19		simpr 110	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → 𝐴 = +∞)
20		qre 9690	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
21	20	rexrd 8069	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ*)
22		pnfxr 8072	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
23		eleq1 2256	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵 ∈ ℝ* ↔ +∞ ∈ ℝ*))
24	22, 23	mpbiri 168	. . . . . . . 8 ⊢ (𝐵 = +∞ → 𝐵 ∈ ℝ*)
25	21, 24	jaoi 717	. . . . . . 7 ⊢ ((𝐵 ∈ ℚ ∨ 𝐵 = +∞) → 𝐵 ∈ ℝ*)
26	25	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → 𝐵 ∈ ℝ*)
27		pnfnlt 9853	. . . . . 6 ⊢ (𝐵 ∈ ℝ* → ¬ +∞ < 𝐵)
28	26, 27	syl 14	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → ¬ +∞ < 𝐵)
29	28	adantr 276	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ +∞ < 𝐵)
30	19, 29	eqnbrtrd 4047	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
31		pnfge 9855	. . . . . . 7 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ +∞)
32	26, 31	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → 𝐵 ≤ +∞)
33	32	adantr 276	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → 𝐵 ≤ +∞)
34	33, 19	breqtrrd 4057	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → 𝐵 ≤ 𝐴)
35	34	notnotd 631	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ ¬ 𝐵 ≤ 𝐴)
36	30, 35	2falsed 703	. 2 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
37		simpl 109	. 2 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 ∈ ℚ ∨ 𝐴 = +∞))
38	18, 36, 37	mpjaodan 799	1 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2164   class class class wbr 4029  ℝcr 7871  +∞cpnf 8051  ℝ^*cxr 8053   < clt 8054   ≤ cle 8055  ℚcq 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