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Theorem axlowdimlem8 28245
Description: Lemma for axlowdim 28257. Calculate the value of 𝑃 at three. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypothesis
Ref Expression
axlowdimlem7.1 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
Assertion
Ref Expression
axlowdimlem8 (𝑃‘3) = -1

Proof of Theorem axlowdimlem8
StepHypRef Expression
1 axlowdimlem7.1 . . 3 𝑃 = ({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))
21fveq1i 6892 . 2 (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3)
3 3ex 12296 . . . 4 3 ∈ V
4 negex 11460 . . . 4 -1 ∈ V
53, 4fnsn 6606 . . 3 {⟨3, -1⟩} Fn {3}
6 c0ex 11210 . . . . 5 0 ∈ V
76fconst 6777 . . . 4 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
8 ffn 6717 . . . 4 ((((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0} → (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}))
97, 8ax-mp 5 . . 3 (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3})
10 disjdif 4471 . . . 4 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
113snid 4664 . . . 4 3 ∈ {3}
1210, 11pm3.2i 471 . . 3 (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})
13 fvun1 6982 . . 3 (({⟨3, -1⟩} Fn {3} ∧ (((1...𝑁) ∖ {3}) × {0}) Fn ((1...𝑁) ∖ {3}) ∧ (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})) → (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3))
145, 9, 12, 13mp3an 1461 . 2 (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3)
153, 4fvsn 7181 . 2 ({⟨3, -1⟩}‘3) = -1
162, 14, 153eqtri 2764 1 (𝑃‘3) = -1
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  cdif 3945  cun 3946  cin 3947  c0 4322  {csn 4628  cop 4634   × cxp 5674   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113  -cneg 11447  3c3 12270  ...cfz 13486
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-sep 5299  ax-nul 5306  ax-pr 5427  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-mulcl 11174  ax-i2m1 11180
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-neg 11449  df-2 12277  df-3 12278
This theorem is referenced by:  axlowdimlem16  28253
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