Theorem ssfzoulel 13123

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 1187	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐴 ∈ ℤ)
2		simpl3 1188	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ 𝐵 ≤ 𝐴) → 𝐵 ∈ ℤ)
3		zre 11977	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
4		zre 11977	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ)
5		ltnle 10712	. . . . . . . . . 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 1125	. . . . . . . 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 13122	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^𝐵) ⊆ (0..^𝑁) → (0 ≤ 𝐴 ∧ 𝐵 ≤ 𝑁)))
10	1, 2, 8, 9	syl3anc 1366	. . . . . 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 10636	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 ∈ ℝ)
13	3	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ)
14		letr 10726	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ 0 ∧ 0 ≤ 𝐴) → 𝐵 ≤ 𝐴))
15	11, 12, 13, 14	syl3anc 1366	. . . . . . . . . . . . . . 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 1125	. . . . . . . . . 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 11898	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
24	4, 23, 3	3anim123i 1146	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
25	24	3coml 1122	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ))
26		letr 10726	. . . . . . . . . . . 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 1082   ∈ wcel 2108   ⊆ wss 3934   class class class wbr 5057  (class class class)co 7148  ℝcr 10528  0cc0 10529   < clt 10667   ≤ cle 10668  ℕ₀cn0 11889  ℤcz 11973  ..^cfzo 13025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026

This theorem is referenced by:  swrdnd2  14009