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Theorem axlowdimlem8 27896
Description: Lemma for axlowdim 27908. 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 6843 . 2 (𝑃‘3) = (({⟨3, -1⟩} ∪ (((1...𝑁) ∖ {3}) × {0}))‘3)
3 3ex 12234 . . . 4 3 ∈ V
4 negex 11398 . . . 4 -1 ∈ V
53, 4fnsn 6559 . . 3 {⟨3, -1⟩} Fn {3}
6 c0ex 11148 . . . . 5 0 ∈ V
76fconst 6728 . . . 4 (((1...𝑁) ∖ {3}) × {0}):((1...𝑁) ∖ {3})⟶{0}
8 ffn 6668 . . . 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 4431 . . . 4 ({3} ∩ ((1...𝑁) ∖ {3})) = ∅
113snid 4622 . . . 4 3 ∈ {3}
1210, 11pm3.2i 471 . . 3 (({3} ∩ ((1...𝑁) ∖ {3})) = ∅ ∧ 3 ∈ {3})
13 fvun1 6932 . . 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 7126 . 2 ({⟨3, -1⟩}‘3) = -1
162, 14, 153eqtri 2768 1 (𝑃‘3) = -1
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  cdif 3907  cun 3908  cin 3909  c0 4282  {csn 4586  cop 4592   × cxp 5631   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7356  0cc0 11050  1c1 11051  -cneg 11385  3c3 12208  ...cfz 13423
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-1cn 11108  ax-icn 11109  ax-addcl 11110  ax-mulcl 11112  ax-i2m1 11118
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7359  df-neg 11387  df-2 12215  df-3 12216
This theorem is referenced by:  axlowdimlem16  27904
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