Theorem ssfzoulel 13336

Description: If a half-open integer range is a subset of a half-open range of nonnegative integers, but its lower bound is greater than or equal to the upper bound of the containing range, or its upper bound is less than or equal to 0, then its upper bound is less than or equal to its lower bound (and therefore it is actually empty). (Contributed by Alexander van der Vekens, 24-May-2018.)

Assertion

Ref	Expression
ssfzoulel	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))

Proof of Theorem ssfzoulel

Step	Hyp	Ref	Expression
1		simpl2 1194	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℤ)
2		simpl3 1195	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℤ)
3		zre 12180	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 12180	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		ltnle 10912	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
6	3, 4, 5	syl2an 599	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
7	6	3adant1 1132	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
8	7	biimpar 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 < 𝐵)
9		ssfzo12 13335	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
10	1, 2, 8, 9	syl3anc 1373	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
11	4	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ)
12		0red 10836	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ)
13	3	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ)
14		letr 10926	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
15	11, 12, 13, 14	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
16	15	expcomd 420	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴)))
17	16	imp 410	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≤ 𝐴) → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴))
18	17	con3d 155	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≤ 𝐴) → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
19	18	ex 416	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
20	19	3adant1 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
21	20	com23 86	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (¬ 𝐵 ≤ 𝐴 → (0 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
22	21	imp 410	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (0 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
23		nn0re 12099	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
24	4, 23, 3	3anim123i 1153	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
25	24	3coml 1129	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
26		letr 10926	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 𝑁 ∧ 𝑁 ≤ 𝐴) → 𝐵 ≤ 𝐴))
27	25, 26	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 𝑁 ∧ 𝑁 ≤ 𝐴) → 𝐵 ≤ 𝐴))
28	27	expdimp 456	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐵 ≤ 𝑁) → (𝑁 ≤ 𝐴 → 𝐵 ≤ 𝐴))
29	28	con3d 155	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐵 ≤ 𝑁) → (¬ 𝐵 ≤ 𝐴 → ¬ 𝑁 ≤ 𝐴))
30	29	impancom 455	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (𝐵 ≤ 𝑁 → ¬ 𝑁 ≤ 𝐴))
31	22, 30	anim12d 612	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁) → (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴)))
32		ioran 984	. . . . . . . 8 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝑁 ≤ 𝐴 ∧ ¬ 𝐵 ≤ 0))
33	32	biancomi 466	. . . . . . 7 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴))
34	31, 33	syl6ibr 255	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁) → ¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)))
35	10, 34	syld 47	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → ¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)))
36	35	con2d 136	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ¬ (𝐴..^𝐵) ⊆ (0..^𝑁)))
37	36	impancom 455	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)) → (¬ 𝐵 ≤ 𝐴 → ¬ (𝐴..^𝐵) ⊆ (0..^𝑁)))
38	37	con4d 115	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0)) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴))
39	38	ex 416	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   ∈ wcel 2110   ⊆ wss 3866   class class class wbr 5053  (class class class)co 7213  ℝcr 10728  0cc0 10729   < clt 10867   ≤ cle 10868  ℕ₀cn0 12090  ℤcz 12176  ..^cfzo 13238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239

This theorem is referenced by:  swrdnd2  14220