Theorem xqltnle 10627

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 10603	. . . 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 9957	. . . . . . 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 10114	. . . . 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 4137	. . . 4 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
9	5	renepnfd 8324	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ≠ +∞)
10	9	neneqd 2433	. . . . . 6 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → ¬ 𝐴 = +∞)
11	5	rexrd 8323	. . . . . . 7 ⊢ (((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 ∈ ℚ) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ*)
12		xgepnf 10149	. . . . . . 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 4127	. . . 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 9957	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
21	20	rexrd 8323	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ*)
22		pnfxr 8326	. . . . . . . . 9 ⊢ +∞ ∈ ℝ*
23		eleq1 2295	. . . . . . . . 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 10120	. . . . . 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 4127	. . 3 ⊢ ((((𝐴 ∈ ℚ ∨ 𝐴 = +∞) ∧ (𝐵 ∈ ℚ ∨ 𝐵 = +∞)) ∧ 𝐴 = +∞) → ¬ 𝐴 < 𝐵)
31		pnfge 10122	. . . . . . 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 4137	. . . 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 2203   class class class wbr 4109  ℝcr 8126  +∞cpnf 8305  ℝ^*cxr 8307   < clt 8308   ≤ cle 8309  ℚcq 9951

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-n0 9497  df-z 9578  df-q 9952  df-rp 9987

This theorem is referenced by:  pcadd2  13039