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Theorem suppssfz 13946
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 8847 . . . . . . . 8 (𝐹 ∈ (𝐵m0) → 𝐹 Fn ℕ0)
42, 3syl 17 . . . . . . 7 (𝜑𝐹 Fn ℕ0)
5 nn0ex 12465 . . . . . . . 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 8143 . . . . 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 5148 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (𝑆 < 𝑥𝑆 < 𝑛))
13 fveqeq2 6890 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((𝐹𝑥) = 𝑍 ↔ (𝐹𝑛) = 𝑍))
1412, 13imbi12d 345 . . . . . . . . . . . 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 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 12468 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
21 nn0re 12468 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℕ0𝑆 ∈ ℝ)
2217, 21syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑆 ∈ ℝ)
23 lenlt 11279 . . . . . . . . . . . . . . . . . . . 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 13579 . . . . . . . . . . . . . . . . . 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 2952 . . . . . . . . . . . . . . . 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 3986 . 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 2941  wral 3062  Vcvv 3475  wss 3946   class class class wbr 5144   Fn wfn 6530  cfv 6535  (class class class)co 7396   supp csupp 8133  m cmap 8808  cr 11096  0cc0 11097   < clt 11235  cle 11236  0cn0 12459  ...cfz 13471
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-supp 8134  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-er 8691  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-n0 12460  df-z 12546  df-uz 12810  df-fz 13472
This theorem is referenced by:  fsuppmapnn0fz  13948  fsfnn0gsumfsffz  19834
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