Theorem xqltnle 10447

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 10423	. . . 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 9781	. . . . . . 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 9938	. . . . 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 4087	. . . 4 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
9	5	renepnfd 8158	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ≠ +∞)
10	9	neneqd 2399	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ 𝐴 = +∞)
11	5	rexrd 8157	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*)
12		xgepnf 9973	. . . . . . 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 675	. . . . 5 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ +∞ ≤ 𝐴)
15	7, 14	eqnbrtrd 4077	. . . 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 800	. 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 9781	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
21	20	rexrd 8157	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ*)
22		pnfxr 8160	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
23		eleq1 2270	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵 ∈ ℝ* ↔ +∞ ∈ ℝ*))
24	22, 23	mpbiri 168	. . . . . . . 8 ⊢ (𝐵 = +∞ → 𝐵 ∈ ℝ*)
25	21, 24	jaoi 718	. . . . . . 7 ⊢ ((𝐵 ∈ ℚ ∨ 𝐵 = +∞) → 𝐵 ∈ ℝ*)
26	25	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → 𝐵 ∈ ℝ*)
27		pnfnlt 9944	. . . . . 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 4077	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
31		pnfge 9946	. . . . . . 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 4087	. . . 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 704	. 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 800	1 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   = wceq 1373   ∈ wcel 2178   class class class wbr 4059  ℝcr 7959  +∞cpnf 8139  ℝ^*cxr 8141   < clt 8142   ≤ cle 8143  ℚcq 9775

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-q 9776  df-rp 9811

This theorem is referenced by:  pcadd2  12779