Theorem xqltnle 10590

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 10566	. . . 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 529	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℚ)
4		qre 9920	. . . . . . 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 10077	. . . . 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 4121	. . . 4 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
9	5	renepnfd 8289	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ≠ +∞)
10	9	neneqd 2424	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ 𝐴 = +∞)
11	5	rexrd 8288	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*)
12		xgepnf 10112	. . . . . . 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 680	. . . . 5 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ +∞ ≤ 𝐴)
15	7, 14	eqnbrtrd 4111	. . . 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 529	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) → (𝐵 ∈ ℚ ∨ 𝐵 = +∞))
18	2, 16, 17	mpjaodan 806	. 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 9920	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
21	20	rexrd 8288	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ*)
22		pnfxr 8291	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
23		eleq1 2294	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵 ∈ ℝ* ↔ +∞ ∈ ℝ*))
24	22, 23	mpbiri 168	. . . . . . . 8 ⊢ (𝐵 = +∞ → 𝐵 ∈ ℝ*)
25	21, 24	jaoi 724	. . . . . . 7 ⊢ ((𝐵 ∈ ℚ ∨ 𝐵 = +∞) → 𝐵 ∈ ℝ*)
26	25	adantl 277	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → 𝐵 ∈ ℝ*)
27		pnfnlt 10083	. . . . . 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 4111	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
31		pnfge 10085	. . . . . . 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 4121	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → 𝐵 ≤ 𝐴)
35	34	notnotd 635	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ ¬ 𝐵 ≤ 𝐴)
36	30, 35	2falsed 710	. 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 806	1 ⊢ (((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2202   class class class wbr 4093  ℝcr 8091  +∞cpnf 8270  ℝ^*cxr 8272   < clt 8273   ≤ cle 8274  ℚcq 9914

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-n0 9462  df-z 9541  df-q 9915  df-rp 9950

This theorem is referenced by:  pcadd2  12994