Theorem xqltnle 10305

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 10283	. . . 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 9662	. . . . . . 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 9818	. . . . 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 4049	. . . 4 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
9	5	renepnfd 8044	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ≠ +∞)
10	9	neneqd 2381	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ 𝐴 = +∞)
11	5	rexrd 8043	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*)
12		xgepnf 9853	. . . . . . 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 4039	. . . 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 9662	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
21	20	rexrd 8043	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ*)
22		pnfxr 8046	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
23		eleq1 2252	. . . . . . . . 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 9824	. . . . . 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 4039	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
31		pnfge 9826	. . . . . . 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 4049	. . . 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 2160   class class class wbr 4021  ℝcr 7845  +∞cpnf 8025  ℝ^*cxr 8027   < clt 8028   ≤ cle 8029  ℚcq 9656

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-po 4317  df-iso 4318  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-pnf 8030  df-mnf 8031  df-xr 8032  df-ltxr 8033  df-le 8034  df-sub 8166  df-neg 8167  df-reap 8568  df-ap 8575  df-div 8666  df-inn 8956  df-n0 9213  df-z 9290  df-q 9657  df-rp 9691

This theorem is referenced by:  pcadd2  12385