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Theorem suppssfz 13958
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 8858 . . . . . . . 8 (𝐹 ∈ (𝐵m0) → 𝐹 Fn ℕ0)
42, 3syl 17 . . . . . . 7 (𝜑𝐹 Fn ℕ0)
5 nn0ex 12477 . . . . . . . 8 0 ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ0 ∈ V)
7 suppssfz.z . . . . . . 7 (𝜑𝑍𝑉)
84, 6, 73jca 1128 . . . . . 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 8155 . . . . 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 5152 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑆 < 𝑥𝑆 < 𝑛))
13 fveqeq2 6900 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑛) = 𝑍))
1412, 13imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝑛 → ((𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) ↔ (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍)))
1514rspcva 3610 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍))
16 simplr 767 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ ℕ0)
17 suppssfz.s . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℕ0)
1817adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → 𝑆 ∈ ℕ0)
1918adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑆 ∈ ℕ0)
20 nn0re 12480 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
21 nn0re 12480 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
2217, 21syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℝ)
23 lenlt 11291 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2420, 22, 23syl2anr 597 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2524biimpar 478 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛𝑆)
26 elfz2nn0 13591 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0𝑆 ∈ ℕ0𝑛𝑆))
2716, 19, 25, 26syl3anbrc 1343 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ (0...𝑆))
2827a1d 25 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆)))
2928ex 413 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ ℕ0) → (¬ 𝑆 < 𝑛 → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆))))
30 eqneqall 2951 . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 413 . . . . . . . . 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 407 . . . . . 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 407 . . . 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 3988 . 2 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
451, 44mpdan 685 1 (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  Vcvv 3474  wss 3948   class class class wbr 5148   Fn wfn 6538  cfv 6543  (class class class)co 7408   supp csupp 8145  m cmap 8819  cr 11108  0cc0 11109   < clt 11247  cle 11248  0cn0 12471  ...cfz 13483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484
This theorem is referenced by:  fsuppmapnn0fz  13960  fsfnn0gsumfsffz  19850
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