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Theorem suppssfz 13908
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 8809 . . . . . . . 8 (𝐹 ∈ (𝐵m0) → 𝐹 Fn ℕ0)
42, 3syl 17 . . . . . . 7 (𝜑𝐹 Fn ℕ0)
5 nn0ex 12427 . . . . . . . 8 0 ∈ V
65a1i 11 . . . . . . 7 (𝜑 → ℕ0 ∈ V)
7 suppssfz.z . . . . . . 7 (𝜑𝑍𝑉)
84, 6, 73jca 1129 . . . . . 6 (𝜑 → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉))
98adantr 482 . . . . 5 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍𝑉))
10 elsuppfn 8106 . . . . 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 5113 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑆 < 𝑥𝑆 < 𝑛))
13 fveqeq2 6855 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑛) = 𝑍))
1412, 13imbi12d 345 . . . . . . . . . . . 12 (𝑥 = 𝑛 → ((𝑆 < 𝑥 → (𝐹𝑥) = 𝑍) ↔ (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍)))
1514rspcva 3581 . . . . . . . . . . 11 ((𝑛 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝑆 < 𝑛 → (𝐹𝑛) = 𝑍))
16 simplr 768 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ ℕ0)
17 suppssfz.s . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℕ0)
1817adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → 𝑆 ∈ ℕ0)
1918adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑆 ∈ ℕ0)
20 nn0re 12430 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
21 nn0re 12430 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
2217, 21syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℝ)
23 lenlt 11241 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2420, 22, 23syl2anr 598 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ0) → (𝑛𝑆 ↔ ¬ 𝑆 < 𝑛))
2524biimpar 479 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛𝑆)
26 elfz2nn0 13541 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0...𝑆) ↔ (𝑛 ∈ ℕ0𝑆 ∈ ℕ0𝑛𝑆))
2716, 19, 25, 26syl3anbrc 1344 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → 𝑛 ∈ (0...𝑆))
2827a1d 25 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 ∈ ℕ0) ∧ ¬ 𝑆 < 𝑛) → ((𝐹𝑛) ≠ 𝑍𝑛 ∈ (0...𝑆)))
2928ex 414 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 414 . . . . . . . . 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 408 . . . . . 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 408 . . . 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 3954 . 2 ((𝜑 ∧ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍)) → (𝐹 supp 𝑍) ⊆ (0...𝑆))
451, 44mpdan 686 1 (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  Vcvv 3447  wss 3914   class class class wbr 5109   Fn wfn 6495  cfv 6500  (class class class)co 7361   supp csupp 8096  m cmap 8771  cr 11058  0cc0 11059   < clt 11197  cle 11198  0cn0 12421  ...cfz 13433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434
This theorem is referenced by:  fsuppmapnn0fz  13910  fsfnn0gsumfsffz  19768
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