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Theorem axlowdimlem8 28207
Description: Lemma for axlowdim 28219. 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 6893 . 2 (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3)
3 3ex 12294 . . . 4 3 ∈ V
4 negex 11458 . . . 4 -1 ∈ V
53, 4fnsn 6607 . . 3 {⟨3, -1⟩} Fn {3}
6 c0ex 11208 . . . . 5 0 ∈ V
76fconst 6778 . . . 4 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
8 ffn 6718 . . . 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 4472 . . . 4 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
113snid 4665 . . . 4 3 ∈ {3}
1210, 11pm3.2i 472 . . 3 (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})
13 fvun1 6983 . . 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 1462 . 2 (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3) = ({⟨3, -1⟩}‘3)
153, 4fvsn 7179 . 2 ({⟨3, -1⟩}‘3) = -1
162, 14, 153eqtri 2765 1 (𝑃‘3) = -1
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  cdif 3946  cun 3947  cin 3948  c0 4323  {csn 4629  cop 4635   × cxp 5675   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  -cneg 11445  3c3 12268  ...cfz 13484
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-sep 5300  ax-nul 5307  ax-pr 5428  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-mulcl 11172  ax-i2m1 11178
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-neg 11447  df-2 12275  df-3 12276
This theorem is referenced by:  axlowdimlem16  28215
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