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Theorem axlowdimlem9 28965
Description: Lemma for axlowdim 28976. Calculate the value of 𝑃 away from three. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypothesis
Ref Expression
axlowdimlem7.1 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
Assertion
Ref Expression
axlowdimlem9 ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃𝐾) = 0)

Proof of Theorem axlowdimlem9
StepHypRef Expression
1 axlowdimlem7.1 . . 3 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21fveq1i 6907 . 2 (𝑃𝐾) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾)
3 eldifsn 4786 . . 3 (𝐾 ∈ ((1...𝑁) ∖ {3}) ↔ (𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3))
4 disjdif 4472 . . . . 5 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
5 3ex 12348 . . . . . . 7 3 ∈ V
6 negex 11506 . . . . . . 7 -1 ∈ V
75, 6fnsn 6624 . . . . . 6 {⟨3, -1⟩} Fn {3}
8 c0ex 11255 . . . . . . . 8 0 ∈ V
98fconst 6794 . . . . . . 7 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
10 ffn 6736 . . . . . . 7 ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}))
119, 10ax-mp 5 . . . . . 6 (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})
12 fvun2 7001 . . . . . 6 (({⟨3, -1⟩} Fn {3} ∧ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) ∧ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 𝐾 ∈ ((1...𝑁) ∖ {3}))) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾))
137, 11, 12mp3an12 1453 . . . . 5 ((({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 𝐾 ∈ ((1...𝑁) ∖ {3})) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾))
144, 13mpan 690 . . . 4 (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = ((((1...𝑁) ∖ {3}) × {0})‘𝐾))
158fvconst2 7224 . . . 4 (𝐾 ∈ ((1...𝑁) ∖ {3}) → ((((1...𝑁) ∖ {3}) × {0})‘𝐾) = 0)
1614, 15eqtrd 2777 . . 3 (𝐾 ∈ ((1...𝑁) ∖ {3}) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0)
173, 16sylbir 235 . 2 ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘𝐾) = 0)
182, 17eqtrid 2789 1 ((𝐾 ∈ (1...𝑁) ∧ 𝐾 ≠ 3) → (𝑃𝐾) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  cdif 3948  cun 3949  cin 3950  c0 4333  {csn 4626  cop 4632   × cxp 5683   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  -cneg 11493  3c3 12322  ...cfz 13547
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-neg 11495  df-2 12329  df-3 12330
This theorem is referenced by:  axlowdimlem16  28972  axlowdimlem17  28973
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