Theorem ssfzoulel 13121

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 1188	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℤ)
2		simpl3 1189	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℤ)
3		zre 11972	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 11972	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		ltnle 10706	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
6	3, 4, 5	syl2an 597	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
7	6	3adant1 1126	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
8	7	biimpar 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 < 𝐵)
9		ssfzo12 13120	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
10	1, 2, 8, 9	syl3anc 1367	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
11	4	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ)
12		0red 10630	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ)
13	3	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ)
14		letr 10720	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
15	11, 12, 13, 14	syl3anc 1367	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
16	15	expcomd 419	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴)))
17	16	imp 409	. . . . . . . . . . . . 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 415	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
20	19	3adant1 1126	. . . . . . . . . 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 409	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (0 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
23		nn0re 11893	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
24	4, 23, 3	3anim123i 1147	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
25	24	3coml 1123	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
26		letr 10720	. . . . . . . . . . . 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 455	. . . . . . . . . 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 454	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (𝐵 ≤ 𝑁 → ¬ 𝑁 ≤ 𝐴))
31	22, 30	anim12d 610	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁) → (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴)))
32		ioran 980	. . . . . . . 8 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝑁 ≤ 𝐴 ∧ ¬ 𝐵 ≤ 0))
33	32	biancomi 465	. . . . . . 7 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴))
34	31, 33	syl6ibr 254	. . . . . 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 454	. . 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 415	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   ∈ wcel 2114   ⊆ wss 3924   class class class wbr 5052  (class class class)co 7142  ℝcr 10522  0cc0 10523   < clt 10661   ≤ cle 10662  ℕ₀cn0 11884  ℤcz 11968  ..^cfzo 13023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447  ax-cnex 10579  ax-resscn 10580  ax-1cn 10581  ax-icn 10582  ax-addcl 10583  ax-addrcl 10584  ax-mulcl 10585  ax-mulrcl 10586  ax-mulcom 10587  ax-addass 10588  ax-mulass 10589  ax-distr 10590  ax-i2m1 10591  ax-1ne0 10592  ax-1rid 10593  ax-rnegex 10594  ax-rrecex 10595  ax-cnre 10596  ax-pre-lttri 10597  ax-pre-lttrn 10598  ax-pre-ltadd 10599  ax-pre-mulgt0 10600

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-pss 3942  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-tp 4558  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5446  df-eprel 5451  df-po 5460  df-so 5461  df-fr 5500  df-we 5502  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-pred 6134  df-ord 6180  df-on 6181  df-lim 6182  df-suc 6183  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7567  df-1st 7675  df-2nd 7676  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10663  df-mnf 10664  df-xr 10665  df-ltxr 10666  df-le 10667  df-sub 10858  df-neg 10859  df-nn 11625  df-n0 11885  df-z 11969  df-uz 12231  df-fz 12883  df-fzo 13024

This theorem is referenced by:  swrdnd2  14002