Theorem ssfzoulel 13730

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 1190	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℤ)
2		simpl3 1191	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℤ)
3		zre 12566	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 12566	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		ltnle 11297	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
6	3, 4, 5	syl2an 594	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
7	6	3adant1 1128	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
8	7	biimpar 476	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 < 𝐵)
9		ssfzo12 13729	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
10	1, 2, 8, 9	syl3anc 1369	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
11	4	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ)
12		0red 11221	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ)
13	3	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ)
14		letr 11312	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
15	11, 12, 13, 14	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
16	15	expcomd 415	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴)))
17	16	imp 405	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≤ 𝐴) → (𝐵 ≤ 0 → 𝐵 ≤ 𝐴))
18	17	con3d 152	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 0 ≤ 𝐴) → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
19	18	ex 411	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 ≤ 𝐴 → (¬ 𝐵 ≤ 𝐴 → ¬ 𝐵 ≤ 0)))
20	19	3adant1 1128	. . . . . . . . . 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 405	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (0 ≤ 𝐴 → ¬ 𝐵 ≤ 0))
23		nn0re 12485	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
24	4, 23, 3	3anim123i 1149	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
25	24	3coml 1125	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
26		letr 11312	. . . . . . . . . . . 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 451	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐵 ≤ 𝑁) → (𝑁 ≤ 𝐴 → 𝐵 ≤ 𝐴))
29	28	con3d 152	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝐵 ≤ 𝑁) → (¬ 𝐵 ≤ 𝐴 → ¬ 𝑁 ≤ 𝐴))
30	29	impancom 450	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → (𝐵 ≤ 𝑁 → ¬ 𝑁 ≤ 𝐴))
31	22, 30	anim12d 607	. . . . . . 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 461	. . . . . . 7 ⊢ (¬ (𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) ↔ (¬ 𝐵 ≤ 0 ∧ ¬ 𝑁 ≤ 𝐴))
34	31, 33	imbitrrdi 251	. . . . . 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 134	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ¬ (𝐴..^𝐵) ⊆ (0..^𝑁)))
37	36	impancom 450	. . 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 411	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑁 ≤ 𝐴 ∨ 𝐵 ≤ 0) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → 𝐵 ≤ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   ∈ wcel 2104   ⊆ wss 3947   class class class wbr 5147  (class class class)co 7411  ℝcr 11111  0cc0 11112   < clt 11252   ≤ cle 11253  ℕ₀cn0 12476  ℤcz 12562  ..^cfzo 13631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632

This theorem is referenced by:  swrdnd2  14609