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Theorem suppssfz 13963
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 8861 . . . . . . . 8 (𝐹 ∈ (𝐵m0) → 𝐹 Fn ℕ0)
42, 3syl 17 . . . . . . 7 (𝜑𝐹 Fn ℕ0)
5 nn0ex 12482 . . . . . . . 8 0 ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ0 ∈ V)
7 suppssfz.z . . . . . . 7 (𝜑𝑍𝑉)
84, 6, 73jca 1126 . . . . . 6 (𝜑 → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉))
98adantr 479 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉))
10 elsuppfn 8158 . . . . 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 5151 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑆 < 𝑥𝑆 < 𝑛))
13 fveqeq2 6899 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑛) = 𝑍))
1412, 13imbi12d 343 . . . . . . . . . . . 12 (𝑥 = 𝑛 → ((𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) ↔ (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍)))
1514rspcva 3609 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍))
16 simplr 765 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ ℕ0)
17 suppssfz.s . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℕ0)
1817adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → 𝑆 ∈ ℕ0)
1918adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑆 ∈ ℕ0)
20 nn0re 12485 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
21 nn0re 12485 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
2217, 21syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℝ)
23 lenlt 11296 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2420, 22, 23syl2anr 595 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2524biimpar 476 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛𝑆)
26 elfz2nn0 13596 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0𝑆 ∈ ℕ0𝑛𝑆))
2716, 19, 25, 26syl3anbrc 1341 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ (0...𝑆))
2827a1d 25 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆)))
2928ex 411 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → (¬ 𝑆 < 𝑛 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
30 eqneqall 2949 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) = 𝑍 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆)))
3130a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → ((𝐹𝑛) = 𝑍 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
3229, 31jad 187 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ ℕ0) → ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
3332com23 86 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ0) → ((𝐹𝑛) ≠ 𝑍 → ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → 𝑛 ∈ (0...𝑆))))
3433ex 411 . . . . . . . . . . . 12 (𝜑 → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → 𝑛 ∈ (0...𝑆)))))
3534com14 96 . . . . . . . . . . 11 ((𝑆 < 𝑛 → (𝐹𝑛) = 𝑍) → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆)))))
3615, 35syl 17 . . . . . . . . . 10 ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆)))))
3736ex 411 . . . . . . . . 9 (𝑛 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆))))))
3837pm2.43a 54 . . . . . . . 8 (𝑛 ∈ ℕ0 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → ((𝐹𝑛) ≠ 𝑍 → (𝜑𝑛 ∈ (0...𝑆)))))
3938com23 86 . . . . . . 7 (𝑛 ∈ ℕ0 → ((𝐹𝑛) ≠ 𝑍 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → (𝜑𝑛 ∈ (0...𝑆)))))
4039imp 405 . . . . . 6 ((𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍) → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → (𝜑𝑛 ∈ (0...𝑆))))
4140com13 88 . . . . 5 (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) → ((𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆))))
4241imp 405 . . . 4 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → ((𝑛 ∈ ℕ0 ∧ (𝐹𝑛) ≠ 𝑍) → 𝑛 ∈ (0...𝑆)))
4311, 42sylbid 239 . . 3 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑛 ∈ (𝐹 supp 𝑍) → 𝑛 ∈ (0...𝑆)))
4443ssrdv 3987 . 2 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
451, 44mpdan 683 1 (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  wne 2938  wral 3059  Vcvv 3472  wss 3947   class class class wbr 5147   Fn wfn 6537  cfv 6542  (class class class)co 7411   supp csupp 8148  m cmap 8822  cr 11111  0cc0 11112   < clt 11252  cle 11253  0cn0 12476  ...cfz 13488
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-rep 5284  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-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  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
This theorem is referenced by:  fsuppmapnn0fz  13965  fsfnn0gsumfsffz  19892
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