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Theorem suppssfz 12750
Description: Condition for a function over the nonnegative integers to have a support contained in a finite set of sequential integers. (Contributed by AV, 9-Oct-2019.)
Hypotheses
Ref Expression
suppssfz.z (𝜑𝑍𝑉)
suppssfz.f (𝜑𝐹 ∈ (𝐵𝑚0))
suppssfz.s (𝜑𝑆 ∈ ℕ0)
suppssfz.b (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍))
Assertion
Ref Expression
suppssfz (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑍
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem suppssfz
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 suppssfz.b . 2 (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍))
2 suppssfz.f . . . . . . . 8 (𝜑𝐹 ∈ (𝐵𝑚0))
3 elmapfn 7840 . . . . . . . 8 (𝐹 ∈ (𝐵𝑚0) → 𝐹 Fn ℕ0)
42, 3syl 17 . . . . . . 7 (𝜑𝐹 Fn ℕ0)
5 nn0ex 11258 . . . . . . . 8 0 ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ0 ∈ V)
7 suppssfz.z . . . . . . 7 (𝜑𝑍𝑉)
84, 6, 73jca 1240 . . . . . 6 (𝜑 → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉))
98adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉))
10 elsuppfn 7263 . . . . 5 ((𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉) → (𝑛 ∈ (𝐹 supp 𝑍) ↔ (𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍)))
119, 10syl 17 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑛 ∈ (𝐹 supp 𝑍) ↔ (𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍)))
12 breq2 4627 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑆 < 𝑥𝑆 < 𝑛))
13 fveq2 6158 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → (𝐹𝑥) = (𝐹𝑛))
1413eqeq1d 2623 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑛) = 𝑍))
1512, 14imbi12d 334 . . . . . . . . . . . 12 (𝑥 = 𝑛 → ((𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) ↔ (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍)))
1615rspcva 3297 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍))
17 simplr 791 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ ℕ0)
18 suppssfz.s . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℕ0)
1918adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → 𝑆 ∈ ℕ0)
2019adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑆 ∈ ℕ0)
21 nn0re 11261 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
22 nn0re 11261 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
2318, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℝ)
24 lenlt 10076 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2521, 23, 24syl2anr 495 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2625biimpar 502 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛𝑆)
27 elfz2nn0 12388 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0𝑆 ∈ ℕ0𝑛𝑆))
2817, 20, 26, 27syl3anbrc 1244 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ (0...𝑆))
2928a1d 25 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆)))
3029ex 450 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → (¬ 𝑆 < 𝑛 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
31 eqneqall 2801 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = 𝑍 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆)))
3231a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → ((𝐹𝑛) = 𝑍 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
3330, 32jad 174 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
3433com23 86 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → ((𝐹𝑛) ≠ 𝑍 → ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → 𝑛 ∈ (0...𝑆))))
3534ex 450 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → 𝑛 ∈ (0...𝑆)))))
3635com14 96 . . . . . . . . . . 11 ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆)))))
3716, 36syl 17 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆)))))
3837ex 450 . . . . . . . . 9 (𝑛 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆))))))
3938pm2.43a 54 . . . . . . . 8 (𝑛 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆)))))
4039com23 86 . . . . . . 7 (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → (𝜑𝑛 ∈ (0...𝑆)))))
4140imp 445 . . . . . 6 ((𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍) → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → (𝜑𝑛 ∈ (0...𝑆))))
4241com13 88 . . . . 5 (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → ((𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆))))
4342imp 445 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → ((𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆)))
4411, 43sylbid 230 . . 3 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑛 ∈ (𝐹 supp 𝑍) → 𝑛 ∈ (0...𝑆)))
4544ssrdv 3594 . 2 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
461, 45mpdan 701 1 (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  Vcvv 3190  wss 3560   class class class wbr 4623   Fn wfn 5852  cfv 5857  (class class class)co 6615   supp csupp 7255  𝑚 cmap 7817  cr 9895  0cc0 9896   < clt 10034  cle 10035  0cn0 11252  ...cfz 12284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285
This theorem is referenced by:  fsuppmapnn0fz  12752  fsfnn0gsumfsffz  18319
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