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Theorem axlowdimlem8 26662
Description: Lemma for axlowdim 26674. 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 6664 . 2 (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3)
3 3ex 11707 . . . 4 3 ∈ V
4 negex 10872 . . . 4 -1 ∈ V
53, 4fnsn 6405 . . 3 {⟨3, -1⟩} Fn {3}
6 c0ex 10623 . . . . 5 0 ∈ V
76fconst 6558 . . . 4 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
8 ffn 6507 . . . 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 4417 . . . 4 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
113snid 4591 . . . 4 3 ∈ {3}
1210, 11pm3.2i 471 . . 3 (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})
13 fvun1 6747 . . 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 1452 . 2 (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3)
153, 4fvsn 6935 . 2 ({⟨3, -1⟩}‘3) = -1
162, 14, 153eqtri 2845 1 (𝑃‘3) = -1
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  cdif 3930  cun 3931  cin 3932  c0 4288  {csn 4557  cop 4563   × cxp 5546   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  -cneg 10859  3c3 11681  ...cfz 12880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-neg 10861  df-2 11688  df-3 11689
This theorem is referenced by:  axlowdimlem16  26670
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